Hypersingular Integrals and Their Applications

Transcription

1 Hypersingular Integrals and Their Applications Stefan G. Samko Rostov State University, Russia and University ofalgarve, Portugal London and New York

2 Contents Preface xv Notation 1 Part 1. Hypersingular integrals 3 Chapter 1. Some basics from the theory of special functions and operator theory 5 1. Some special functions Euler gamma- and beta-functions Orthogonal Gegenbauer and Chebyshev polynomials C^(t), A > 0, andt m (t) Bessel functions J v {z) t I v (z) and K v [z) The Gauss hypergeometric function 9 2. Basics of the theory of spherical harmonics The basis of spherical harmonics The addition theorem and the Funk-Hekke formula The connection between the smoothness of a function and the convergence of its Fourier-Laplace series On fractional integration and differentiation of functions of one variable Fourier transforms and the Wiener ring; Fourier multipliers The Wiener ring Fourier multipliers Miscellaneous On estimation of the Taylor remainder for Holderian functions Analyticity of integrals depending on a parameter ' Hadamard's approach to divergent integrals and their regularization Estimates of some multidimensional integrals Bibliographical notes to Chapter 1 34 Chapter 2. The Riesz potential operator and Lizorkin type invariant spaces $v The Riesz potential operator Definitions and the Fourier transform of the Riesz potential in the case 0 < 3?a < a^i Mapping properties of the operator I a in the spaces L p (R n ) and L p (R";p) The Lizorkin space $ 39

3 viii CONTENTS 3. The Riesz potential in the general case 5fta > 0 and invariance of the space $ The function l/\x\ a as a distribution in S'(R n ) or $'(R n ) The Fourier transform of the distribution l/ sc a The Riesz kernel and the Riesz potential in the general case Lizorkin-type spaces $v Definitions, examples and simple properties Denseness of *v and $y in L p (R n ) Connections of the Riesz potential with the Poisson and Gauss-Weierstrass semigroups of operators Bibliographical notes to Chapter 2 54 Chapter 3. Hypersingular integrals with constant characteristics The Riesz fractional differentiation W Fourier transform of the hypersingular integral W f Zeros of the function d ni i(a) The phenomenon of annihilation of hypersingular integrals Convergence of the hypersingular integral W f in the case of smooth functions; moments of hypersingular integrals Diminution of order I > Sfta to I > 2 [^p] in the case of a non-centred difference The hypersingular operator W as inverse to the Riesz potential operator I a Auxiliary functions Ai io,(x,h), ki <a (x) and * fl,( x ) The Inversion theorem Almost everywhere convergence of the Riesz derivative Reducing the order a: the case of complex a More on the independence of W f on the order of finite differences A Zygmund type estimate for the continuity modulus of hypersingular integrals Connection of hypersingular integrals with fractional powers of operators Hypersingular integrals with generalized differences Generalized differences Modified hypersingular integrals and their Fourier transforms Properties of the function At{a) Modified hypersingular integral as inverse to the Riesz potential operator Bibliographical notes to Chapter 3 84 Chapter 4. Potentials and hypersingular integrals with homogeneous characteristics Preliminaries on some means of functions on the unit sphere in R n Some means of functions on the unit sphere On the choice of a rotation which is smooth with respect to the parameter On smoothness of the means M u,(x',t) Cavalieri type principle and the means of spherical harmonics Potential type operators with homogeneous characteristics The symbol in the case 0 < 3ta < 1 95

4 CONTENTS ix 2.2. The Hadamard constructions of divergent integrals over the sphere in the case of singularity at an equator On the symbol K^{x) in the case 3ta > Smoothness properties of the symbol IC"(x) On the inversion of the operator "characteristic symbol" Hypersingular integrals with homogeneous characteristics Classification of hypersingular integrals Symbol of a hypersingular integral A hypersingular integral as a convolution with the function "iwifl Symbol of a hypersingular operator as the Fourier transform of a distribution Some estimates of hypersingular integrals of smooth functions vanishing at infinity Smoothness properties of the symbol X>^(x) Representation of homogeneous differential operators in partial derivatives by means of hypersingular operators Hypersingular integrals with harmonic characteristics On a certain integral equation of the first kind on the unit sphere: the case of a non-integer a The case of integer a and the main representation theorem Bibliographical notes to Chapter Chapter 5. Hypersingular integrals with non-homogeneous characteristics On the connection Wft ic f = AJD?f The limiting operator A = lirrie-vo A e and justification of the convergence Integrability of the kernel a(x) and its Fourier transform Justification of the convergence A e > A as e > The main theorems on the convergence of E3^ ef Representation of a hypersingular integral as a convolution with a function of the type. Ay o and the inverse problem On a /.p.-convolution Representation of a hypersingular integral as a /.p.-convolution On a functional equation On the inverse problem On the continuity modulus of a hypersingular integral Bibliographical notes to Chapter Chapter 6. Hypersingular integrals on the unit sphere Spherical convolution operators. Operators commuting with rotations Spherical convolution operators on harmonics Operators commuting with rotations Generalized functions (distributions) on the sphere Boundedness of convolution operators in L p (S n ~ 1 ) Spherical potentials The spherical Riesz potential operator Connection with the spatial Riesz potential Other potentials Spherical hypersingular integrals and inversion of the spherical Riesz potential 159

5 x CONTENTS 3.1. Spherical hypersingular integrals of order 0 < 3ta < The multiplier of the spherical hypersingular operator Justification of the inversion in Z^S" 1 " 1 ); identity approximation on the sphere Inversion of the spherical Riesz potential in the case 3ta > Zygmund type estimates for spherical potential operators and spherical hypersingular integrals The continuity modulus of functions on the sphere Technical lemma The case of potentials The case of hypersingular integrals Bibliographical notes to Chapter Part 2. Applications of hypersingular integrals 173 Chapter 7. Characterization of some function spaces in terms of hypersingular integrals The spaces I a (L p ) of Riesz potentials Definitions and local behaviour of Riesz potentials Characterization of the space, I a (L p ) in terms of convergence of hypersingular integrals The spaces B a (L p ) of Bessel potentials Denseness of Cfi (R n ) in the space L^(R n ) Proof of the main result B<*(L P ) = L^{R") The spaces L^r(R n ) 187 r 3.1. Connection with the space of Riesz potentials Imbeddings in the spaces L^r(R n ) Weak derivatives of functions fel^r(r n ) Riesz derivatives W*f of integer order and powers of the Laplace operator, fei a {L p ) The spaces Wjf r and the characterization of the space I a (L p ) in terms of higher derivatives Spaces L pr (p p,p r ) with Muckenhoupt weight functions Simultaneous approximation of functions and their Riesz derivatives in different L p -norms Convergence in L p of hypersingular integrals with non-stabilizing characteristics Hypersingular integrals and differences of fractional order Fractional differentiation in a given direction The case of ordinary fractional differences The case of mixed fractional differences The case of operator fractional difference Uniform estimates Spaces of Riesz potentials with densities in Orlicz spaces Definitions On boundedness of the Riesz potential operator in Orlicz spaces The space I a (E M ) of Riesz potentials The spaces E^^RT) Characterization of the spaces L"(5 n ~ 1 ) 223

6 CONTENTS 8.1. Characterization of the range. a (L p ) Proof of the coincidence & a {L p ) = L^S"" 1 ) Bibliographical notes to Chapter Chapter 8. Solution of multidimensional integral equations of the first kind with a potential type kernel Inversion of potential type operators with homogeneous characteristics in the elliptic case General remarks on the inversion Structure of the Fourier transform of the reciprocal of the symbol of a potential Representation of F~ x ( %?) by an f.p.-integral over the sphere The associated characteristics Inversion of potentials of non-integer order a by hypersingular operators Inversion of potentials of even order a = 2,4,6,... by hypersingular operators Inversion of potentials of odd order a = 1,3,5,... by hypersingular operators On the inversion of potentials of integer order in the general case Extension to the case of densities in L p (R n ) The annihilation of the kernel of a potential by the associated hyper- ' singular operator Fundamental solutions of hypersingular operators Integral representation of truncated hypersingular integrals The Fourier transform of the kernel of the representation The kernel of the representation as an identity approximation kernel in the case of associated characteristics Convergence in L p of the truncated hypersingular integrals and inversion of potentials within the framework of L p -spaces Inversion of potentials with homogeneous harmonic characteristics The general case The case of a linear characteristic Realization of the inversion for densities in L p (R n ) in the case of a linear characteristic Inversion of potentials with homogeneous characteristics in the case of linear degeneracy of the symbol Characteristics of potentials with a linear degeneracy of the symbol Inversion in the case of "nice" functions The inversion in the case of functions (p L p Inversion of potentials with radial difference characteristics What is to be done? The class C m "{R\) On a Hankel transformation The analytical continuation of the Fourier transform of the function z a (l + :c 2 )-* Construction of the inverse operator 265

7 xii CONTENTS 5.6. Hypersingular integrals as inverses: the cases 0 < a < 1 and 1< a < Justification of the inversion in the Lizorkin space $(i? n ) Justification of the inversion in the space L p : the cases 0 < a < 1 and 1< a < Bibliographical notes to Chapter Chapter 9. Hypersingular operators as positive fractional powers of some operators of mathematical physics Inversion of the Bessel potentials The idea of the construction Properties of the function w_ a (r) The operator (/ A)"/ 2 inverse to the Bessel potential operator Justification of the inversion in the case of "nice" functions The case 0 < 3?a < Justification of the inversion in L p : the cases 0 < 3?a < 1 and 1< So < Another form for (/-A) 01 / Another expression for (/ - A)"/ Parabolic (heat) hypersingular integrals Parabolic fractional potentials Positive fractional powers (Jj - A x )" /2 and (/ + - A x )" /2, a > Justification of the inversion in the case of "nice" functions The case of functions in L p The spaces of parabolic potentials Fractional powers of the hyperbolic (wave) operator The Riesz hyperbolic potential in the case n = :2. The hyperbolic hypersingular integral: the case n = Hyperbolic hypersingular integrals: the general case n > Fractional powers of the Schrodinger operator The fractional Schrodinger potential operator Analytical continuation of the Schrodinger fractional potential Positive fractional powers of the Schrodinger operator Fractional powers of some other differential operators Bibliographical notes to Chapter Chapter 10. Regularization of multidimensional integral equations of the first kind with a potential type kernel Regularization of the equations in it" The case 0 < a < The case o > Application of spherical hypersingular operators to regularization of integral equations of the first kind on the unit sphere Bibliographical notes to Chapter Chapter 11. Some modifications of hypersingular integrals and their applications The case of the Riesz potential operator 316

8 CONTENTS xiii 1.1. General requirements for the kernel q a (y) Convergence in (11.4) on "nice" functions Construction (11.4) as the inverse operator to I a on I"(L p ) The approximative inverse operator under the choice k(x) = P(x, 1) The approximative inverse operator under the direct choice of q a (x) Inversion of potential type operators with homogeneous characteristics by approximative inverses General requirements for the kernel q a {x) in (11.4) Approximative inverses under the choice K{ ) = e~^ Inversion in the non-elliptic case Inversion of Bessel potentials by approximative inverses Approximative inverses under the choice IC e ( ) = e~ e^ The approximative inverses under the choice e ( ) = * a Inversion of the acoustic potentials by approximative inverses Some preliminaries The choice of approximative inverses The representation for (A a )~ 1 A a,c> The inversion theorem Inversion of potentials with oscillating symbols by approximative inverses The kernel corresponding to the oscillating symbol B ai -y( ) Passage to the symbol The inversion Other applications of the approximative inverses method Bibliographical notes to Chapter References 343 Author index 355 Subject index 357 Index of Symbols 359