Journal of Pseudo-Differential Operators and Applications最新文献

In this present paper, we concern investigating nonlinear Kirchhoff-type problems subject to Dirichlet boundary conditions, incorporating nonlocal terms and logarithmic nonlinearity in the (phi )-Hilfer fractional spaces with the (eta (cdot ))-Laplacian operator by means of the do Mountain Pass Theorem, Fountain Theorem and Dual Fountain Theorem.

This article marks the inaugural exploration of certain classes of negative or positive definite functions in several p-adic variables. Associated with these functions, two classes of pseudo-differential operators, convolution semigroups, some positive measures and (L^{2}(mathbb {Q}_{p}^{n}))-sub-Markovian semigroups are introduced.

It is well-known that the pseudodifferential operator with the symbol in Bony class, a subset of (S_{1,1}^0(mathbb R^n)), is bounded on (L^{2}(mathbb {R}^{n})). The main purpose of this paper is to extend the classical results to multi-parameter case, i.e., to discuss the boundedness on (L^2(mathbb {R}^{n_1+n_2})) and on (h^{p}(mathbb {R}^{n_{1}}times mathbb {R}^{n_{2}}) (0<ple 1)) of multi-parameter pseudodifferential operator with symbol satisfying multi-parameter Bony conditions.

The aim of this paper is to establish the boundedness of the fractional type Marcinkiewicz integral operator (mathcal {M}_{alpha ,rho ,m}) and its higher order commutator (mathcal {M}_{alpha ,rho ,m,b^l}) generated by (bin textrm{BMO}({mathbb {R}}^n)) and (mathcal {M}_{alpha ,rho ,m}) on the weighted Lebesgue spaces (L_omega ^p({mathbb {R}}^n)). Under assumption that the variable exponents (alpha (cdot )) and (q(cdot )) satisfy the (log ) decay at infinity and origin, the authors show that the (mathcal {M}_{alpha ,rho ,m}) and (mathcal {M}_{alpha ,rho ,m,b^l}) are bounded on the grand variable Herz spaces (dot{K}_{q(cdot )}^{alpha (cdot ),p),theta }({mathbb {R}}^n)) and the grand variable Herz-Morrey spaces (Mdot{K}_{p),theta ,q(cdot )}^{alpha (cdot ),lambda }({mathbb {R}}^n)), respectively.

In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel (alpha )-potentials. These operators are denoted and defined in the form

where f is a p-adic distribution and (left[ max {1,|varvec{phi }(||zeta ||_{p})|}right] ^{-alpha }) is the symbol of the operator. We will study some properties of the convolution kernel (denoted as (K_{alpha })) of the pseudo-differential operator (mathcal {E}_{varvec{phi },alpha }), (alpha in mathbb {R}); and demonstrate that the family ((K_{alpha })_{alpha >0}) determines a convolution semigroup on (mathbb {Q}_{p}^{n}). Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.

This paper explores a one-parameter extension of the Hartley kernel expressed as a real combination of two Bessel functions, termed the Hartley–Bessel function. The key feature of the Hartley–Bessel function is derived through a limit transition from the (-1) little Jacobi polynomials. The Hartley–Bessel function emerges as an eigenfunction of a first-order difference-differential operator and possesses a Sonin integral-type representation. Our main contribution lies in investigating anovel product formula for this function, which subsequently facilitates the development of innovative generalized translation and convolution structures on the real line. The obtained product formula is expressed as an integral in terms of this function with an explicit non-positive and uniformly bounded measure. Consequently, a non-positivity-preserving convolution structure is established.

A short proof of M. W. Wong’s inequality (leftVert J_{-s}varphi rightVert _p le varepsilon leftVert J_{-t}varphi rightVert _p + C leftVert varphi rightVert _p) is given.

In this paper, for the Dirac operator and three One-forms we give the proof of the another general Dabrowski–Sitarz–Zalecki type theorem for the spectral Einstein functional on odd dimensional manifolds with boundary.

In this paper, we study the existence of solutions for the following class of Euclidean bosonic equations with Liouville–Weyl fractional derivatives

where (beta in (0,frac{1}{2})), ({_{-infty }}D_{x}^{beta }u(cdot ), {_{x}}D_{infty }^{beta }u(cdot )) denote the left and right Liouville–Weyl fractional derivatives, (omega ,Q:{mathbb {R}}rightarrow {mathbb {R}}) is a positive function with (omega ,Qin L^{frac{1}{2beta }} ({mathbb {R}})) and (g: {mathbb {R}}rightarrow {mathbb {R}}) is a continuous function satisfying suitable conditions. Finally, an example is provided.

Consider the differential operator

where I is the identity operator. The operator (Delta _{a,b}) is known as affine Laplacian. We consider the heat equation associated to the operator (Delta _{a,b}) with initial condition f from (L^2({mathbb {R}}^n)). Its solution is denoted by (e^{tDelta _{a,b}}f). The transform (f mapsto e^{tDelta _{a,b}}f) is called affine heat kernel transform (or A-heat kernel transform). In this article, we consider (analytically extended) affine heat kernel transform and characterize the image of (displaystyle L^2({mathbb {R}})) under it as a weighted Bergman space of analytic functions on ({mathbb {C}}) with nonnegative weight. Consequently, we study (L^p)-boundedness of affine heat kernel transform, (L^p)-boundedness of affine Bargmann projection and related duality results. Moreover, we define affine Weyl translations and characterize the maximal and minimal spaces of analytic functions on ({mathbb {C}}) which are invariant under the affine Weyl translations.