Journal of Pseudo-Differential Operators and Applications最新文献

In this paper, we acquire the boundedness of commutators generated by bilinear Calderón–Zygmund operator and (text {BMO}) functions on two weighted Herz spaces with variable exponents.

Bi-g-frame, was introduced as a pair of operator sequences, could obtain a new reconstruction formula for elements in Hilbert spaces. In this paper we aim at studying the characterizations and constructions of bi-g-frames. For a bi-g-frame ((Lambda ,,Gamma )), the relationship between the sequence (Lambda ) and the sequence (Gamma ) is very crucial, we are devoted to characterizing bi-g-frames, whose component the sequences are g-Bessel sequences, g-frames and so on. Then we discuss the construction of new bi-g-frames, we show that bi-g-frames can be constructed by specific operators, dual g-frames and g-dual frames. Especially, we also study those bi-g-frames for which one of the constituent sequences is a g-orthonormal basis.

In this article, we discuss subspace duals of a frame of translates by an action of a closed abelian subgroup (Gamma ) of a locally compact group ({mathscr {G}}.) These subspace duals are not required to lie in the space generated by the frame. We characterise translation-generated subspace duals of a frame/Riesz basis involving the Zak transform for the pair (({mathscr {G}}, Gamma ).) We continue our discussion on the orthogonality of two translation-generated Bessel pairs using the Zak transform, which allows us to explore the dual of super-frames. As an example, we extend our findings to splines, Gabor systems, p-adic fields ({mathbb {Q}} p,) locally compact abelian groups using the fiberization map.

Abstract

We study first-order symmetrizable hyperbolic (Ntimes N) systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at (x=0) , these systems take the form $$begin{aligned} partial _t u + {{mathcal {A}}}(t,x,y,xD_x,D_y) u = f(t,x,y), quad (t,x,y)in (0,T)times {{mathbb {R}}}_+times {{mathbb {R}}}^d, end{aligned}$$ where ({{mathcal {A}}}(t,x,y,xD_x,D_y)) is a first-order differential operator with coefficients smooth up to (x=0) and the derivative with respect to x appears in the combination (xD_x) . No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator (partial _t + {{mathcal {A}}}(t,x,y,xD_x,D_y)) is well-posed in that scale. More specifically, solutions u exhibit formal asymptotic expansions of the form $$begin{aligned} u(t,x,y) sim sum _{(p,k)} frac{(-1)^k}{k!}x^{-p} log ^k !x , u_{pk}(t,y) quad hbox { as} xrightarrow +0 end{aligned}$$ where ((p,k)in {{mathbb {C}}}times {{mathbb {N}}}_0) and (Re prightarrow -infty ) as (|p|rightarrow infty ) , provided that the right-hand side f and the initial data (u|_{t=0}) admit asymptotic expansions as (x rightarrow +0) of a similar form, with the singular exponents p and their multiplicities unchanged. In fact, the coefficients (u_{pk}) are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients  (u_{pk}) solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator (partial _t+{{mathcal {A}}}(t,x,y,xD_x,D_y)) is well-posed in the scale of standard Sobolev spaces (H^s((0,T)times {{mathbb {R}}}_+^{1+d})) .

We prove in this paper a generalization of Heisenberg inequality for Gabor transform in the setup of the semidirect product (mathbb {R}^nrtimes K), where K is a compact subgroup of automorphisms of (mathbb {R}^n). We also solve the sharpness problem and thus we obtain an optimal analogue of the Heisenberg inequality. A local uncertainty inequality for the Gabor transform is also provided, in the same context. This allows us to prove a couple of global uncertainty inequalities. The representation theory and Plancherel formula are fundamental tools in the proof of our results.

In this paper, we study the unique continuation property for the fractional p-elliptic equations in a semigroup form with variable coefficients. By employing an extension procedure, we derive a monotonicity formula for an extended frequency function. Utilizing this monotonicity together with a blow-up analysis, we establish the unique continuation property.

First, we introduce the concept of triple sequence space (c^3(triangle )) and we define a Hausdorff measure of noncompactness (MNC) on this space. Furthermore, by using this MNC we study the existence of solutions of infinite systems of Caputo–Hadamard fractional differential equations with three point integral boundary conditions in the triple sequence space ( c^3(triangle )). Finally, we give an example to show the effectiveness of our main result.

In analogy with the definition of “extended Sobolev scale" on (mathbb {R}^n) by Mikhailets and Murach, working in the setting of the lattice (mathbb {Z}^n), we define the “extended Sobolev scale" (H^{varphi }(mathbb {Z}^n)), where (varphi ) is a function which is RO-varying at infinity. Using the scale (H^{varphi }(mathbb {Z}^n)), we describe all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of discrete Sobolev spaces ([H^{(s_0)}(mathbb {Z}^n), H^{(s_1)}(mathbb {Z}^n)]), with (s_0<s_1). We use this interpolation result to obtain the mapping property and the Fredholmness property of (discrete) pseudo-differential operators (PDOs) in the context of the scale (H^{varphi }(mathbb {Z}^n)). Furthermore, starting from a first-order positive-definite (discrete) PDO A of elliptic type, we define the “extended discrete A-scale" (H^{varphi }_{A}(mathbb {Z}^n)) and show that it coincides, up to norm equivalence, with the scale (H^{varphi }(mathbb {Z}^n)). Additionally, we establish the (mathbb {Z}^n)-analogues of several other properties of the scale (H^{varphi }(mathbb {R}^n)).

In this paper, the offset linear canonical transform associated with biquaternion is defined, which is called the biquaternion offset linear canonical transforms (BiQOLCT). Then, the inverse transform and Plancherel formula of the BiQOLCT are obtained. Next, Heisenberg uncertainty principle and Donoho-Stark’s uncertainty principle for the BiQOLCT are established. Finally, as an application, we study signal recovery by using Donoho-Stark’s uncertainty principle associated with the BiQOLCT.

Abstract

In this paper, making use of non-smooth variational principle, we establish the existence of solution to the following Kirchhoff type mixed local and nonlocal elliptic problem with concave–convex and Choquard nonlinearities $$begin{aligned} left{ begin{array}{ll} mathcal {L}_{a,b}(u)=left( int limits _{Omega }frac{|u(y)|^{p}}{|x-y|^{mu }}dyright) |u(x)|^{p-2}u(x)+lambda |u(x)|^{q-2}u(x), &{}quad xin Omega , ~~~u(x)ge 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{}quad xin Omega , ~u(x)=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{}quad xin mathbb {R}^{N}setminus Omega , end{array} right. end{aligned}$$ where (mathcal {L}_{a,b}(u)=-left( a+b Vert nabla uVert ^{2(gamma -1)}_{L^{2}(Omega )}right) Delta u(x)+(-Delta )^s u(x)) , (gamma in left( 1,frac{N+4s+2}{N-2}right) ) , (a>0) , (b>0) are constants, ((-Delta )^{s}) is the restricted fractional Laplacian, (0<s<1) , (1<q<2<2p) , (0<mu <N) . The main contribution of this paper is giving a new supercritical range of (2p-1) and (gamma ) .