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The main objective of this paper is to enrich the theoretical system of the linear canonical Fourier transform (LCFT) by introducing the canonical potential J m , A and corresponding L p -Sobolev space. Moreover, the Schwartz-type space S A is introduced. Further, pseudo-differential operator (PDO) is defined and obtained its another integral representation. The L p -boundedness result for the pseudo-differential operator associated with the LCFT is discussed. Some applications of Sobolev-type spaces H A m , p and H A m , p are given. [ABSTRACT FROM AUTHOR]

The pseudo-differential operator involving the Kontorovich–Lebedev transform is defined. The boundedness of pseudo differential operator in a certain Sobolev-type space is shown. Composition of pseudo-differential operatorsis defined in terms of the two symbols, and also its boundedness in certain Sobolev-type space is given. Some special cases are also discussed. Finally, the Kontorovich–Lebedev transform is used in the solution of some partial differential equations. [ABSTRACT FROM PUBLISHER]

We give an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev–Dunkl spaces. Next, we come up with some results regarding the identity operator, the Dunkl transform and pseudo-differential operators. [ABSTRACT FROM AUTHOR]

We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points. [ABSTRACT FROM AUTHOR]

In this article, we study the effect of the geometry of a domain with variable local dimension on the regularity/singularity of the restriction of a multifractional random field on such a domain. The theories of reproducing kernel Hilbert spaces (RKHS) and generalized random fields are applied. Fractional Sobolev spaces of variable order are considered as RKHSs of random fields satisfying certain elliptic multifractional pseudodifferential equations. The multifractal spectra of these random fields are trivial due to the regularity assumptions on the variable order of the fractional derivatives. In this article, we introduce a family of RKHSs defined by isomorphic identification with the trace on a compact heterogeneous fractal domain of a fractional Sobolev space of variable order. The local regularity/singularity order of functions in these spaces, which depends on the variable order of the fractional Sobolev space considered and on the local dimension of the domain, is derived. We also study the spectral properties of the family of models introduced in the mean-square sense. In the Gaussian case, random fields with sample paths having multifractional local Hölder exponent are covered in this framework. [ABSTRACT FROM PUBLISHER]

Let [image omitted] be an open set, and [image omitted] where [image omitted] We consider pseudodifferential operators in domain Ω+ with double symbols which have singularities near [image omitted] and super exponential growths at infinity. We suppose that symbols have analytic extension with respect to the variable dual to the time in the lower complex half-plane. We construct the theory of invertibility of such operators in weighted Sobolev spaces with weights connected with growths of symbols. We give applications to estimates of the fundamental solutions of such operators, in particular, to the heat equations with singular potentials of power, exponential and super exponential growths. [ABSTRACT FROM AUTHOR]

A class of pseudo-differential operators (p.d.o.'s) associated with the general Fourier kernel studied by Hardy and Titchmarsh is defined. A symbol class Tm is introduced. It is shown that the p.d.o.'s associated with the symbol are continuous linear mappings of the Braaksma and Schuitman space T(λ,μ) into itself. An integral representation of p.d.o. is obtained. Some special forms of the symbol are considered. It is shown that these p.d.o.'s and their products are bounded in certain Sobolev type space. [ABSTRACT FROM AUTHOR]

An approximation Ansatz for the operator solution, U ( z ′, z ), of a hyperbolic first-order pseudodifferential equation, ∂̸ z + a ( z , x , D x ) with Re ( a ) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L ( H ( s ) , H ( s ) ) of these operators is provided, which yields a convergence result for the Ansatz to U ( z ′, z ) in some Sobolev space as the number of operators in the composition goes to ∞. [ABSTRACT FROM AUTHOR]

We study the class of random fields having their reproducing kernel Hilbert space isomorphic to a fractional Sobolev space of variable order on Rn. Prototypes of this class include multifractional Brownian motion, multifractional free Markov fields, and multifractional Riesz-Bessel motion. The study is carried out using the theory of generalized random fields defined on fractional Sobolev spaces of variable order. Specifically, we consider the class of generalized random fields satisfying a pseudoduality condition of variable order. The factorization of the covariance operator of the pseudodual allows the definition of a white-noise linear filter representation of variable order. In the ordinary case, the Holder continuity, in the mean-square sense, of the class of random fields introduced is proved, and its mean-square Holder spectrum is defined in terms of the variable regularity order of the functions in the associated reproducing kernel Hilbert space. The pseudodifferential representation of variable order of the resulting class of multifractal random fields is also defined. Some examples of pseudodifferential models of variable order are then given. [ABSTRACT FROM AUTHOR]