Integral Transformations, Operational Calculus and Their Applications Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Hari Mohan Srivastava Edited by Integral Transformations, Operational Calculus and Their Applications Integral Transformations, Operational Calculus and Their Applications Editor Hari Mohan Srivastava MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Hari Mohan Srivastava University of Victoria Canada Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) (available at: https://www.mdpi.com/journal/symmetry/special issues/Integral Transformations Operational Calculus Their Applications). 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Contents About the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Hari Mohan Srivastava Special Issue of Symmetry: “Integral Transformations,Operational Calculus and Their Applications” Reprinted from: Symmetry 2020 , 12 , 1169, doi:10.3390/sym12071169 . . . . . . . . . . . . . . . . . 1 Ron Kerman Construction of Weights for Positive Integral Operators Reprinted from: Symmetry 2020 , 12 , 1004, doi:10.3390/sym12061004 . . . . . . . . . . . . . . . . . 5 Slavko Simi ́ c and Bandar Bin-Mohsin Some Improvements of the Hermite–Hadamard Integral Inequality Reprinted from: Symmetry 2020 , 12 , 117, doi:10.3390/sym12010117 . . . . . . . . . . . . . . . . . 21 Nak Eun Cho, Mohamed Kamal Aouf and Rekha Srivastava The Principle of Differential Subordination and Its Application to Analytic and p -Valent Functions Defined by a Generalized Fractional Differintegral Operator Reprinted from: Symmetry 2019 , 11 , 1083, doi:10.3390/sym11091083 . . . . . . . . . . . . . . . . 33 Lei Shi, Mohsan Raza, Kashif Javed, Saqib Hussain and Muhammad Arif Class of Analytic Functions Defined by q -Integral Operator in a Symmetric Region Reprinted from: Symmetry 2019 , 11 , 1042, doi:10.3390/sym11081042 . . . . . . . . . . . . . . . . 47 Rabha W. Ibrahim and Maslina Darus New Symmetric Differential and Integral Operators Defined in the Complex Domain Reprinted from: Symmetry 2019 , 11 , 906, doi:10.3390/sym11070906 . . . . . . . . . . . . . . . . . 63 Shahid Mahmood, Gautam Srivastava, Hari Mohan Srivastava, Eman S. A. Abujarad, Muhammad Arif and Fazal Ghani Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points Reprinted from: Symmetry 2018 , 11 , 764, doi:10.3390/sym11060764 . . . . . . . . . . . . . . . . . 75 Eman S. A. AbuJarad, Nusrat Raza, Shahid Mahmood, Gautam Srivastava, H. M. Srivastava, and Sarfraz Nawaz Malik Geometric Properties of Certain Classes of Analytic Functions Associated with a q -Integral Operator Reprinted from: Symmetry 2019 , 11 , 719, doi:10.3390/sym11050719 . . . . . . . . . . . . . . . . . 83 Hari M. Srivastava, Anupam Das, Bipan Hazarika and S. A. Mohiuddine Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra Reprinted from: Symmetry 2019 , 11 , 674, doi:10.3390/sym11050674 . . . . . . . . . . . . . . . . . 97 Ndolane Sene and Gautam Srivastava Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations Reprinted from: Symmetry 2019 , 11 , 608, doi:10.3390/sym11050608 . . . . . . . . . . . . . . . . . 113 v Lei Shi, Hari Mohan Srivastava, Muhammad Arif, Shehzad Hussain and Hassan Khan An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function Reprinted from: Symmetry 2019 , 11 , 598, doi:10.3390/sym11050598 . . . . . . . . . . . . . . . . . 125 Mostafa Bachar On Periodic Solutions of Delay Differential Equations with Impulses Reprinted from: Symmetry 2019 , 11 , 523, doi:10.3390/sym11040523 . . . . . . . . . . . . . . . . . 139 Tianyu Huang, Xijuan Guo, Yue Zhang, Zheng Chang Collaborative Content Downloading in VANETs with Fuzzy Comprehensive Evaluation Reprinted from: Symmetry 2019 , 11 , 502, doi:10.3390/sym11040502 . . . . . . . . . . . . . . . . . 151 Hari Mohan Srivastava, Bidu Bhusan Jena, Susanta Kumar Paikray and Umakanta Misra Statistically and Relatively ModularDeferred-Weighted Summability and Korovkin-Type Approximation Theorems Reprinted from: Symmetry 2019 , 11 , 448, doi:10.3390/sym11040448 . . . . . . . . . . . . . . . . . 169 Hari M. Srivastava, Faruk ̈ Ozger and S. A. Mohiuddine Construction of Stancu-Type Bernstein Operators Based on B ́ ezier Bases with Shape Parameter λ Reprinted from: Symmetry 2019 , 11 , 316, doi:10.3390/sym11030316 . . . . . . . . . . . . . . . . . 189 vi About the Editor Hari Mohan Srivastava was born on 5 July, 1940 in Karon (District Ballia) in the Province of Uttar Pradesh in India. Professor Hari Mohan Srivastava began his university-level teaching career right after having received his M.S. degree in Mathematics in 1959 at the age of 19. He earned his Ph.D. degree in 1965 while he was a full-time member of the teaching faculty at the Jai Narain Vyas University of Jodhpur in India (since 1963). Currently, Professor Srivastava holds the position of Professor Emeritus in the Department of Mathematics and Statistics at the University of Victoria in Canada, having joined the faculty in 1969. Professor Srivastava has held (and continues to hold) numerous Visiting and Chair Professorships at many universities and research institutes in different parts of the world. Having received several D.S. (honoris causa) degrees as well as honorary memberships and fellowships of many scientific academies and scientific societies around the world, he is also actively editorially associated with numerous international scientific research journals as an Honorary or Advisory Editor or as an Editorial Board Member. He has also edited (and is currently editing) many Special Issues of scientific research journals as the Lead or Joint Guest Editor, including (for example) the MDPI journals Axioms, Mathematics, and Symmetry; the Elsevier journals Journal of Computational and Applied Mathematics, Applied Mathematics and Computation, Chaos, Solitons & Fractals, Alexandria Engineering Journal, and Journal of King Saud University—Science, the Wiley journal, Mathematical Methods in Applied Sciences; the Springer journals Advances in Difference Equations, Journal of Inequalities and Applications, Fixed Point Theory and Applications, and Boundary Value Problems; the American Institute of Physics journal Chaos: An Interdisciplinary Journal of Nonlinear Science; the American Institute of Mathematical Sciences journal AIMS Mathematics; the Hindawi journals Advances in Mathematical Physics, International Journal of Mathematics and Mathematical Sciences, and Abstract and Applied Analysis; the De Gruyter (now the Tbilisi Centre for Mathematical Sciences) journal Tbilisi Mathematical Journal; the Yokohama Publisher journal Journal of Nonlinear and Convex Analysis; the University of Nis journal Filomat; the Ministry of Communications and High Technologies (Republic of Azerbaijan) journal Applied and Computational Mathematics: An International Journal, and so on. He is a Clarivate Analytics (Thomson Reuters, Web of Science) Highly-Cited Researcher. Professor Srivastava’s research interests include several areas of pure and applied mathematical sciences, such as real and complex analysis, fractional calculus and its applications, integral equations and transforms, higher transcendental functions and their applications, q-series and q-polynomials, analytic number theory, analytic and geometric inequalities, probability and statistics, and inventory modeling and optimization. He has published 36 books, monographs, and edited volumes, 36 book (and encyclopedia) chapters, 48 papers in international conference proceedings, and more than 1350 peer-reviewed international scientific research journal articles, as well as forewords and prefaces to many books and journals. Further details about Professor Srivastava’s professional achievements and scholarly accomplishments, as well as honors, awards, and distinctions, can be found at the following web site: http://www.math.uvic.ca/ harimsri/. vii symmetry S S Editorial Special Issue of Symmetry: “Integral Transformations, Operational Calculus and Their Applications” Hari Mohan Srivastava 1,2,3 1 Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada; harimsri@math.uvic.ca 2 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 3 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan Received: 7 July 2020; Accepted: 7 July 2020; Published: 14 July 2020 This Special Issue consists of a total of 14 accepted submissions (including several invited feature articles) to the Special Issue of the MDPI’s journal, Symmetry on the general subject-area of “Integral Transformations, Operational Calculus and Their Applications” from different parts of the world. The present Special Issue contains the invited, accepted and published submissions (see [ 1 – 14 ]) to a Special Issue of the MDPI’s journal, Symmetry , on the remarkably wide subject-area of “Integral Transformations, Operational Calculus and Their Applications”. Many successful predecessors of this Special Issue happen to be the Special Issues of the MDPI’s journal, Axioms , on the subject-areas of “ q -Series and Related Topics in Special Functions and Analytic Number Theory”, “Mathematical Analysis and Applications" and “Mathematical Analysis and Applications II", the Special Issues of Mathematics , on the subject-areas of “Recent Advances in Fractional Calculus and Its Applications”, “Recent Developments in the Theory and Applications of Fractional Calculus”, “Operators of Fractional Calculus and Their Applications” and “Fractional-Order Integral and Derivative Operators and Their Applications", and indeed also the Special Issue of Symmetry itself, on the subject-area of “Integral Transforms and Operational Calculus”. In fact, encouraged by the noteworthy successes of this series of Special Issues, as well as of (for example) the two Special Issues of Axioms , on the subject-areas of “Mathematical Analysis and Applications” and “Mathematical Analysis and Applications II”, Axioms has already started the publication of a Topical Collection, entitled “Mathematical Analysis and Applications” (Collection Editor: H. M. Srivastava), with an open submission deadline. The interested reader should refer to and read the book format of many of these Special Issues (Guest Editor: H. M. Srivastava), which are cited below (see [15–18]). In recent years, various families of fractional-order integral and derivative operators, such as those named after Riemann-Liouville, Weyl, Hadamard, Grünwald-Letnikov, Riesz, Erdélyi-Kober, Liouville-Caputo and so on, have been found to be remarkably important and fruitful, due mainly to their demonstrated applications in numerous seemingly diverse and widespread areas of the mathematical, physical, chemical, engineering and statistical sciences. Many of these fractional-order operators provide interesting and potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral and integro-differential equations; fractional-calculus analogues and extensions of each of these equations; and various other problems involving special functions of mathematical physics and applied mathematics, as well as their extensions and generalizations in one or more variables (see, for details, a widely- and extensively-cited monograph [19]). As it is known fairly well, investigations involving the theory and applications of integral transformations and operational calculus are remarkably wide-spread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences. In this Special Issue, we invited and welcome review, expository and original research articles dealing with the recent Symmetry 2020 , 12 , 1169; doi:10.3390/sym12071169 www.mdpi.com/journal/symmetry 1 Symmetry 2020 , 12 , 1169 state-of-the-art advances on the topics of integral transformations and operational calculus as well as their multidisciplinary applications, together with some relevance to the aspect of symmetry. The suggested topics of interest for the call of papers for this Special Issue included, but were not limited to, the following keywords: • Integral Transformations and Integral Equations as well as Other Related Operators Including Their Symmetry Properties and Characteristics • Applications Involving Mathematical (or Higher Transcendental) Functions Including Their Symmetry Properties and Characteristics • Applications Involving Fractional-Order Differential and Differintegral Equations and Their Associated Symmetry • Applications Involving Symmetrical Aspect of Geometric Function Theory of Complex Analysis • Applications Involving q -Series and q -Polynomials and Their Associated Symmetry • Applications Involving Special Functions of Mathematical Physics and Applied Mathematics and Their Symmetrical Aspect • Applications Involving Analytic Number Theory and Symmetry Several well-established scientific research journals, which are published by such publishers as (for example) Elsevier Science Publishers, John Wiley and Sons, Hindawi Publishing Corporation, Springer, De Gruyter, MDPI and other publishing houses, have published and continue to publish a number of Special Issues of many of their journals on recent advances on different aspects, especially of the subject of one of the above-mentioned keywords, “Applications Involving Fractional-Order Differential and Differintegral Equations”. Many widely-attended international conferences, too, continue to be successfully organized and held world-wide ever since the very first one on this particular subject-area in U.S.A. in the year 1974. In this connection, it seems to be worthwhile to refer the interested readers of this Special Issue to a recently-published survey-cum-expository review article (see [ 20 ]) which presented a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional “differintegral” equations. Furthermore, in connection with such works as (for example) [ 4 , 7 ], and indeed also many papers included in the published volumes (see [ 15 – 18 ]), a recent survey-cum-expository review article (see [21]) will be potentially useful in order to motivate further researches and developments involving a wide variety of operators of basic (or q -) calculus and fractional q -calculus and their widespread applications in Geometric Function Theory of Complex Analysis. In the same survey-cum-expository review article (see [ 21 ]), it is also pointed out as to how known results for the q -calculus can easily (and possibly trivially) be translated into the corresponding results for the so-called ( p , q ) -calculus (with 0 < q < p ≤ 1) by applying some obvious parametric and argument variations, the additional parameter p being redundant (or superfluous). Finally, I take this opportunity to thank all of the participating authors, and the referees and the peer-reviewers, for their invaluable contributions toward the remarkable success of each of the above-mentioned Special Issues. I do also greatly appreciate the editorial and managerial help and assistance provided efficiently and generously by Mr. Philip Li, Ms. Linda Cui and Ms. Grace Wang, and also many of their colleagues and associates in the Editorial Office of Symmetry Conflicts of Interest: The author declares no conflict of interest. 2 Symmetry 2020 , 12 , 1169 References 1. Kerman, R. Construction of Weights for Positive Integral Operators. Symmetry 2020 , 12 , 1004. [CrossRef] 2. Simi ́ c, S.; Bin-Mohsin, B. Some Improvements of the Hermite-Hadamard Integral Inequality. Symmetry 2020 , 12 , 117. [CrossRef] 3. Cho, N.E.; Aouf, M.K.; Srivastava, R. The Principle of Differential Subordination and Its Application to Analytic and p -Valent Functions Defined by a Generalized Fractional Differintegral Operator. Symmetry 2019 , 11 , 1083. [CrossRef] 4. Shi, L.; Raza, M.; Javed, K.; Hussain, S.; Arif, M. Class of Analytic Functions Defined by q -Integral Operator in a Symmetric Region. Symmetry 2019 , 11 , 1042. [CrossRef] 5. Ibrahim, R.W.; Darus, M. New Symmetric Differential and Integral Operators Defined in the Complex Domain. Symmetry 2019 , 11 , 906. [CrossRef] 6. Mahmood, S.; Srivastava, G.; Srivastava, H.M.; Abujarad, E.S.A.; Arif, M.; Ghani, F. Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points. Symmetry 2019 , 11 , 764. [CrossRef] 7. Mahmood, S.; Raza, N.; AbuJarad, E.S.A.; Srivastava, G.; Srivastava, H.M.; Malik, S.N. Geometric Properties of Certain Classes of Analytic Functions Associated with a q -Integral Operator. Symmetry 2019 , 11 , 719. [CrossRef] 8. Srivastava, H.M.; Das, A.; Hazarika, B.; Mohiuddine, S.A. Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra. Symmetry 2019 , 11 , 674. [CrossRef] 9. Sene, N.; Srivastava, G. Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations. Symmetry 2019 , 11 , 608. [CrossRef] 10. Shi, L.; Srivastava, H.M.; Arif, M.; Hussain, S.; Khan, H. An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function. Symmetry 2019 , 11 , 598. [CrossRef] 11. Bachar, M. On Periodic Solutions of Delay Differential Equations with Impulses. Symmetry 2019 , 11 , 523. [CrossRef] 12. Huang, T.-Y.; Guo, X.-J.; Zhang, Y.; Chang, Z. Collaborative Content Downloading in VANETs with Fuzzy Comprehensive Evaluation. Symmetry 2019 , 11 , 502. [CrossRef] 13. Srivastava, H.M.; Jena, B.B.; Paikray, S.K.; Misra, U.K. Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems. Symmetry 2019 , 11 , 448. [CrossRef] 14. Srivastava, H.M.; Özger, F.; Mohiuddine, S.A. Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ Symmetry 2019 , 11 , 316. [CrossRef] 15. Srivastava, H.M. Mathematical Analysis and Applications ; (viii + 209 pp.); Printed Edition of the Special Issue “Mathematical Analysis and Applications”; Published in Axioms ; MDPI Publishers: Basel, Switzerland, 2019; ISBN 978-3-03897-400-0 (Pbk); ISBN 978-3-03897-401-7 (PDF). 16. Srivastava, H.M. Integral Transforms and Operational Calculus ; Printed Edition of the Special Issue “Integral Transforms and Operational Calculus”; Published in Symmetry ; MDPI Publishers: Basel, Switzerland, 2019; ISBN 978-3-03921-618-5 (Pbk); ISBN 978-3-03921-619-2 (PDF). 17. Srivastava, H.M. Operators of Fractional Calculus and Their Applications ; (viii + 125 pp.); Printed Edition of the Special Issue “Operators of Fractional Calculus and Their Applications”; Published in Mathematics ; MDPI Publishers: Basel, Switzerland, 2019; ISBN 978-3-03897-340-9 (Pbk); ISBN 978-3-03897-341-6 (PDF). 18. Srivastava, H.M. Mathematical Analysis and Applications II ; (viii + 215 pp.); Printed Edition of the Special Issue “Mathematical Analysis and Applications II”; Published in Axioms ; MDPI Publishers: Basel, Switzerland, 2020; ISBN 978-3-03928-384-2 (Pbk); ISBN 978-3-03928-385-9 (PDF). 19. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations ; (xvi + 523 pp.); North-Holland Mathematical Studies, Vol. 204; Elsevier (North-Holland) Science Publishers: Amsterdam, The Netherlands, 2006; ISBN-10: 0-444-51832-0. 3 Symmetry 2020 , 12 , 1169 20. Srivastava, H.M. Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments. Kyungpook Math. J. 2020 , 60 , 73–116. [CrossRef] 21. Srivastava, H.M. Operators of Basic (or q -) Calculus and Fractional q -Calculus and Their Applications in Geometric Function Theory of Complex Analysis. Iran. J. Sci. Technol. Trans. A Sci. 2020 , 44 , 327–344. [CrossRef] c © 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 4 symmetry S S Article Construction of Weights for Positive Integral Operators Ron Kerman Department of Mathematics, Brock University, St. Catharines, ON L2S 3A1, Canada; rkerman@brocku.ca Received: 15 March 2020; Accepted: 16 April 2020; Published: 12 June 2020 Abstract: Let ( X , M , μ ) be a σ -finite measure space and denote by P ( X ) the μ -measurable functions f : X → [ 0, ∞ ] , f < ∞ μ ae. Suppose K : X × X → [ 0, ∞ ) is μ × μ -measurable and define the mutually transposed operators T and T ′ on P ( X ) by ( T f )( x ) = ∫ X K ( x , y ) f ( y ) d μ ( y ) and ( T ′ g )( y ) = ∫ X K ( x , y ) g ( x ) d μ ( x ) , f , g ∈ P ( X ) , x , y ∈ X Our interest is in inequalities involving a fixed (weight) function w ∈ P ( X ) and an index p ∈ ( 1, ∞ ) such that: (*): ∫ X [ w ( x )( T f )( x )] p d μ ( x ) C ∫ X [ w ( y ) f ( y )] p d μ ( y ) The constant C > 1 is to be independent of f ∈ P ( X ) . We wish to construct all w for which (*) holds. Considerations concerning Schur’s Lemma ensure that every such w is within constant multiples of expressions of the form φ 1/ p − 1 1 φ 1/ p 2 , where φ 1 , φ 2 ∈ P ( X ) satisfy T φ 1 ≤ C 1 φ 1 and T ′ φ 2 ≤ C 2 φ 2 Our fundamental result shows that the φ 1 and φ 2 above are within constant multiples of (**): ψ 1 + ∑ ∞ j = 1 E − j T ( j ) ψ 1 and ψ 2 + ∑ ∞ j = 1 E − j T ′ ( j ) ψ 2 respectively; here ψ 1 , ψ 2 ∈ P ( X ) , E > 1 and T ( j ) , T ′ ( j ) are the j th iterates of T and T ′ This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels K ( x , y ) = K ( y , x ) , so that T ′ = T . This means that only the first series in (**) needs to be studied. Keywords: weights; positive integral operators; convolution operators MSC: 2000 Primary 47B34; Secondary 27D10 1. Introduction Consider a σ -finite measure space ( X , M , μ ) and a positive integral operator T defined through a nonnegative kernel K = K ( x , y ) which is μ × μ measurable on X × X ; that is, T is given on the class, P ( X ) , of μ -measurable functions f : X → [ 0, ∞ ] , f < ∞ μ ae, by ( T f )( x ) = ∫ X K ( x , y ) f ( y ) d μ ( y ) , x ∈ X The transpose, T ′ , of T at g ∈ P ( X ) is ( T ′ g )( y ) = ∫ X K ( x , y ) g ( x ) d μ ( x ) , y ∈ X ; it satisfies ∫ X gT f d μ = ∫ X f T ′ g d μ , f , g ∈ P ( X ) Our focus will be on inequalities of the form ∫ X [ uT f ] p d μ ≤ B p ∫ X [ v f ] p d μ , (1) Symmetry 2020 , 12 , 1004; doi:10.3390/sym12061004 www.mdpi.com/journal/symmetry 5 Symmetry 2020 , 12 , 1004 with the index p fixed in ( 1, ∞ ) and B > 0 independent of f ∈ P ( X ) ; here, u , v ∈ P ( X ) , 0 ≤ u , v < ∞ , μ ae , are so-called weights. The equivalence need only be proved in one direction. Suppose, then, (1) holds and g ∈ P ( x ) satisfies ∫ X [ u − 1 g ] p d μ < ∞ . Then [ ∫ X [ v − 1 T ′ g ] p ′ d μ ] 1 p ′ = sup ∫ X f v − 1 T ′ g d μ , the supremum being take over f ∈ P ( X ) with ∫ X f p d μ ≤ 1. But, Fubini’s Theorem ensures ∫ X f v − 1 T ′ g d μ = ∫ X gT ( f v − 1 ) d μ = ∫ X ( u − 1 g ) uT ( f v − 1 ) d μ ≤ [ ∫ X [ u − 1 g ] p ′ d μ ] 1 p ′ [ B p ∫ X [ v f v − 1 ] p d μ ] 1 p ≤ [ B p ′ ∫ X [ u − 1 g ] p ′ d μ ] 1 p ′ Further, (1) holds if and only if the dual inequality ∫ X [ v − 1 T ′ g ] p ′ d μ ≤ B p ′ ∫ X [ u − 1 g ] p ′ d μ , p ′ = p p − 1 , (2) does. Inequality (1) has been studied for various operators T in such papers as [1–9]. In this paper, we are interested in constructing weights u and v for which (1) holds. We restrict attention the case u = v = w ; the general case will be investigated in the future. Our approach is based on the observation that, implicit in a proof of the converse of Schur’s lemma, given in [ 10 ], is a method for constructing w . An interesting application of Schur’s lemma itself to weighted norm inequalities is given in Christ [11]. In Section 2, we prove a number of general results the first of which is the following one. Theorem 1. Let ( X , M , μ ) be a σ -finite measure space with u , v ∈ P ( X ) , 0 ≤ u , v < ∞ , μ ae Suppose that T is a positive integral operator on P ( X ) with transpose T ′ Then, for fixed p , 1 < p < ∞ , one has (1) , with C > 1 independent of f ∈ P ( X ) , if and only if them exists a function φ ∈ P ( X ) and a constant C > 1 for which T ( v − 1 φ p ′ ) ≤ Cu − 1 φ p ′ and T ′ ( u φ p ) ≤ Cv φ p (3) In this case, B 0 , the smallest B possible in (1) and Co , the smallest possible C so that (3) holds for some φ , satisfy B 0 ≤ C 0 = max [ B p 1 , B p ′ 1 ] , where B 1 = B 1/ p 0 + B 1/ p ′ 0 Theorem 1 has the following consequence. Corollary 1. Under the condition of Theorem 1, (1) holds for u = v = w if and only if w = φ − 1/ p ′ 1 φ 1/ p 2 , where φ 1 , φ 2 are functions in P ( X ) satisfying T φ 1 ≤ C φ 1 and T ′ φ 2 ≤ C φ 2 , (4) 6 Symmetry 2020 , 12 , 1004 for some C > 1 Though it is often possible to work with the inequalities (4) directly (see Remark 1) it is important to have a general method to construct the functions φ 1 and φ 2 . This method is given in our principal result. Theorem 2. Suppose X , μ and T are as in Theorem 1. Let φ ∈ P ( X ) Then, φ satisfies an inequality of the form T φ ≤ C 1 φ , C 1 > 0 constant, (5) if and only if there is a constant C > 1 such that C − 1 φ ≤ ψ + ∞ ∑ j = 1 C − j 2 T ( j ) ψ ≤ C φ , (6) where ψ ∈ P ( X ) , C 2 > 1 is constant and T ( j ) = T ◦ T · · · ◦ T , j times. The kernels of operator of the form ∞ ∑ j = 1 C − j T ( j ) and ∞ ∑ j = 1 C − j T ′ ( j ) will be called the weight generating kernels of T . In Sections 3–6 these kernels will be calculated for particular T . All but the Hardy operators considered in Section 6 operate on the class P ( R n ) of nonnegative, Lebesgue-measurable functions on R n The operators last referred to are, in fact, convolution operators ( T k f )( x ) = ( k ∗ f )( x ) = ∫ R n k ( x − y ) f ( y ) dy , x ∈ R n , with even integrable kernels k , ∫ R n k ( y ) dy = 1. In particular, the kernel k ( x − y ) is symmetric, so T ′ k = T k , whence only the first series in (**) need be considered. Further, the convolution kernels are part of an approximate identity { k t } t > 0 on L P ( R n ) = { f Leb. meas: [ ∫ R n | f | p ] 1/ p < ∞ } , see [ 12 ]. Thus, it becomes of interest to characterize the weights w for which { k t } l > 0 is an approximate identity on L p ( w ) = L p ( R n , w ) = { f Leb. meas: ‖ f ‖ p , w = [ ∫ R n | w f | p ] 1/ p < ∞ } ; that is k t ∗ f ∈ L p ( w ) and lim t → 0 + ‖ k t ∗ f − f ‖ p , w = 0 for all f ∈ L p ( w ) . It is a consequence of the Banach-Steinhaus Theorem that this will be so if and only if sup 0 < t < a ‖ k t ‖ < ∞ 7 Symmetry 2020 , 12 , 1004 for some fixed a > 0, where ‖ k t ‖ denotes the operator norm of T k t on L p ( w ) . We remark here that the operators in Sections 3–5 are bounded on L p ( w ) and, indeed, form part of an approximate identity on L p ( w ) , if w satisfies the A p condition, namely, sup [ 1 | Q | ∫ Q w p ] [ 1 | Q | ∫ Q w − p ′ ] 1/ p ′ < ∞ , p ′ = p p − 1 , (7) the supremum being taken over all cubes Q in R n whose sides are parallel to the coordinate axes with ∞ > | Q | = Lebesgue measure of Q . See ([13], p. 62) and [14]. Finally, all the convolution operators are part of a convolution semigroup ( k t ) t > 0 ; that is k t ( x ) = t − n k ( x t ) and k t 1 ∗ k t 2 = k t 1 + t 2 , t 1 , t 2 > 0. The approximate identity result can thus be interpreted as the continuity of the semigroup. We conclude the introduction with some remarks on terminology and notation. The fact that T is bounded on L p ( w ) if and only if T ′ is bounded on L p ′ ( w − 1 ) is called the principle of duality or, simply, duality. Two functions f , g ∈ P ( X ) are said to be equivalent if a constant C > 1 exists for which C − 1 g ≤ f ≤ Cg (8) We indicate this by f ≈ g , with the understanding that C is independent of all parameters appearing, (except dimension) unless otherwise stated. If only one of the inequalities in (8) holds, we use the notation f g or f g , as appropriate. Lastly, a convolution operator and its kernel are frequently denoted by the same symbol. 2. General Results In this section we give the proofs of the results stated in the Introduction, together with some remarks. Proof of Theorem 1. The conditions (3) are, respectively, equivalent to T ′ : L 1 ( u − 1 φ p ′ ) → L 1 ( v − 1 φ p ′ ) i.e., T : L ∞ ( v φ − p ′ ) → L ∞ ( u φ − p ′ ) and T : L 1 ( v φ p ) → L 1 ( u φ p ) It will suffice to deal with the first condition in (3). So, Fubini’s Theorem yields ∫ X v − 1 φ p ′ T ′ f d μ ≤ C ∫ X u − 1 φ p ′ f d μ equivalent to ∫ X f T ( v − 1 φ p ′ ) d μ ≤ C ∫ X f u − 1 φ p ′ d μ , f ∈ P ( X ) , and hence to T ( v − 1 φ p ′ ) ≤ Cu − 1 φ p ′ , since f is arbitrary. According to the main result of [15], then, T : L p ( ( v φ p ) 1/ p ( v φ − p ′ ) 1/ p ′ ) → L p ( ( u φ p ) 1/ p ( u φ − p ′ ) 1/ p ′ ) i.e., T : L p ( v ) → L p ( u ) , with norm ≤ C , so that (1) holds with B ≤ C 8 Symmetry 2020 , 12 , 1004 Suppose now (1) holds. Following [10], choose g ∈ P ( X ) with ∫ X g pp ′ d μ = 1. Let T 1 g = [ uT ( v − 1 g p ′ ) ] 1/ p ′ and T 2 g = [ v − 1 T ′ ( ug p ) ] 1/ p . Set S = T 1 + T 2 , A = B 0 + ε and φ = ∞ ∑ j = 0 A − j S ( j ) g As in [ 10 ], conclude T 1 φ ≤ A φ and T 2 φ ≤ A φ , so that (2) is satisfied for C 0 ≤ [ B p 1 , B p ′ 1 ] , where B 1 = B 1/ p 0 + B 1/ p ′ 0 Proof of Corollary 1. Given (1) , one has (2) and Theorem 1 then implies (3) , with T replaced by T ′ , namely for u = v = w , T ( w − 1 φ p ′ ) ≤ Cw − 1 φ p ′ and T ( w φ p ) ≤ Cw φ p , whence the inequalities (4) are satisfied by φ 1 = w φ p and φ 2 = w − 1 ψ p ′ . Conversely, given (4) , and taking u = v = w = φ 1/ p − 1 1 φ 1/ p 2 , one readily obtains (3), with ψ = ( ψ 1 ψ 2 ) 1/ pp ′ Proof of Theorem 2. Clearly, if (6) holds, T φ ≤ C [ T ψ + ∞ ∑ j = 1 C − j 2 T ( j + 1 ) ψ ] = CC 2 ∞ ∑ j = 1 C − j 2 T ( j ) ψ ≤ C 2 C 2 φ Suppose φ ∈ P ( X ) satisfies (5). Then, T ( j ) φ ≤ C j 1 φ 1 , j = 1, 2, . . . . It only remains to observe that ( 1 + C 1 ε ) − 1 φ ≤ φ + ∞ ∑ j = 1 ( C 1 + ε ) − j T ( j ) φ ≤ φ + ∞ ∑ j = 1 ( C 1 C 1 + ε ) j φ ≤ ( 1 + C 1 ε ) φ , for any ε > 0. Remark 1. The class of functions φ determined by the weight-generating operators ∞ ∑ j = 1 C − j T ( j ) effectively remains the same as C increases. Thus, suppose 0 < C 1 < C 2 , ψ ∈ P ( X ) and φ = ψ + ∞ ∑ j = 1 C − j 1 T ( j ) ψ Then, φ is equivalent to ψ + ∞ ∑ j = 1 C − j 2 T ( j ) ψ , since φ ≤ φ + ∞ ∑ j = 1 C − j 2 T ( j ) φ = ∞ ∑ j = 0 C − j 2 ∞ ∑ k = 0 C − k 1 T ( j + k ) ψ = ∞ ∑ l = 0 ( ∑ j + h = l C − k 1 C − j 2 ) T ( l ) ψ = ∞ ∑ l = 0 ∞ ∑ j = 0 ( C 1 C 2 ) j C − l 1 T ( l ) ψ = C 2 C 2 − C 1 ∞ ∑ l = 0 C − l 1 T ( l ) ψ = C 2 C 2 − C 1 φ This means that in dealing with weight-generating operators we need only consider C > 1. 9 Symmetry 2020 , 12 , 1004 We conclude this section with the following observations on approximate identities in weighted Lebesgue spaces. Remark 2. Suppose { k t } t > 0 is an approximate identity in L p ( R n ) , 1 < p < ∞ If the inequalities (4) involving φ 1 and φ 2 can be shown to hold for T kt , t ∈ ( 0, a ] for some a > 0, with C > 1 independent of such t , then { k t } t > 0 will also be an approximate identity in L p ( w ) = L p ( R n , w ) , w = φ − 1/ p ′ 1 φ 1/ p 2 Example 1. Let k = k ( | x | ) be any bounded, nonnegative radial function on R n which is a decreasing function of | x | and suppose ∫ R n k ( x ) dx = 1. It is well-known, see ([ 13 ], p. 63), that k t ( x ) = t − n k ( x / t ) , x ∈ R n , is an approximate identity in L p ( R n ) , 1 < p < ∞ The weight w ( x ) = 1 + | x | − n / p ( 1 + log + ( 1 / | x | )) − 1 , for fixed p , 1 < p < ∞ , has the interesting properly that T k t : L p ( w ) → L p ( w ) for all t > 0, yet { k t } t > 0 is never an approximate identity in L p ( w ) To obtain the boundedness assertion take φ 1 ( x ) = 1 and φ 2 ( x ) = 1 + | x | − n ( 1 + log + ( 1/ | x | )) − p in Corollary 1. Arguments similar to those in [ 6 ] show that if { k t } t > 0 is an approximate identity in L p ( w ) , then w must satisfy the A p condition for all cubes Q will sides parallel to the coordinate axes and | Q | ≤ a for some a > 0. However, the weight w does not have this property. 3. The Poisson Integral Operators We recall that for t > 0 and y ∈ R n , the Poisson kernel, P t , is defined by P t ( y ) = c n t ( t 2 + | y | 2 ) − ( n + 1 ) /2 , c n = π − ( n + 1 ) /2 Γ (( n + 1 ) /2 ) Theorem 3. The weight-generating kernels for P t , t > 0, are equivalent to P ≡ P 0 Indeed, given ψ ∈ P ( R n ) , with P ψ < ∞ a.e., C − 1 t P ψ ≤ ∞ ∑ j = 1 C − j P jt ψ ≤ C ′ t P ψ , (9) where C > 1, C t = C max [ t − 1 , t n ] and C ′ t = C t ∞ ∑ j = 1 C − j max [ jt , ( jt ) − n ] Proof. Observe that by the semigroup property P ( j ) t = P jt , j = 1, 2, . . . . Also, min [ t , t − n ] P ≤ P t ≤ max [ t , t − n ] P Now, suppose ψ + ∞ ∑ j = 1 C − j P jt ψ is in P ( R n ) , with C > 1. Then, P ψ ≤ C t P t ψ + ∞ ∑ j = 1 C − j P ( j + 1 ) t ψ ≤ C t ∞ ∑ j = 1 C − j P jt ψ ≤ C t ∞ ∑ j = 1 C − j max [ jt , ( j ) − n ] P ψ ≤ C ′ t P ψ As stated in Section 1, w ∈ A p is sufficient for { P t } t > 0 to be an approximate identify in L P ( w ) Moreover, w ∈ A p is also necessary for this in the periodic case. See [ 6 , 8 , 16 ]. It is perhaps surprising then that the class of approximate identity weights is much larger than A p , as is seen in the next result. 10 Symmetry 2020 , 12 , 1004 Proposition 1. Let w α ( x ) = [ 1 + | x | ] α , α ∈ R Then, for any t > 0, P t is bounded on L p ( w α ) if any only if − n p − 1 < α < n p ′ + 1. Moreover, on that range of α one has lim t → 0 + ‖ P t ∗ f − f ‖ p , ω α = 0, (10) for all f ∈ L p ( ω α ) The set of α for which w α ∈ A p , however, is − n p < α < n p ′ Proof. We omit the easy proof of the assertion concerning the α for which w α ∈ A p To obtain the “if” part of the other assertion we will show P t ∗ w β ≤ Cw β , t > 0, (11) if and only if − n − 1 ≤ β < 1, with C > 1 independent of both s and t , if t ∈ ( 0, 1 ) . Corollary 1 and Remark 2, then yield (10) when − n p − 1 < α < n p ′ + 1. Consider, then, fixed x ∈ R n and 0 < t < 1. We have ( P t ∗ w β )( x ) = ( ∫ | y |≤ | x | 2 + ∫ | x | 2 < | y | < 2 | x | + ∫ | y |≥ 2 | x | ) P t ( y ) w β ( s − t ) dy = I 1 + I 2 + I 3 Now, I 1 ≤ w β ( x ) ∫ | y | < | x | 2 P t ( y ) dy ≤ Cw β ( x ) , for all β ∈ R Again, I 2 ≥ cP t ( x ) ∫ | x − y |≤ 1 ( 1 + | x − y | ) β dy ≥ cP t ( x ) ≥ c | x | − n − 1 , so we require β > n − 1, if (11) is to hold. Moreover, for x ∈ R n and 0 < t < 1, I 2 ≈ P t ( x ) [ | x | n χ | x |≤ 1 + | x | β + n χ | x | > 1 ] ≈ ( | x | t ) n χ | x |≤ 1 + t | x | χ t ≤| x |≤ 1 + t | x | | x | β χ | x |≥ 1 ≤ Cw β ( x ) Next, for | x | • 1 I 3 = ∫ | y | > 2 | x | P t ( y ) w β ( y ) dy t ∫ | y | > 2 | x | | y | − n − 1 + β dy which requires β < 1 to have I 3 < ∞ . In that case I 3 ∫ r > 2 | x | r − n − 1 + β r n − 1 dr | x | β − 1 w β ( x ) That P t is not bounded on L p ( w α ) when α ≤ − n p − 1 can be seen by noting that, for appropriate ε > 0, the function f ( x ) = | x | [ log ( 1 + | x | )] − ( 1 + ε ) / p is in L p ( w α ) , while P t f ≡ ∞ . The range α ≥ n / p + 1 is then ruled out by duality. 11

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