On the constants for multiplication in Sobolev spaces

In the frame work of Sobolev (Bessel potential) spaces Hn(Rd, R or C), we consider the nonlinear Nemytskij operator sending a function x ∈ Rd ↦ f(x) into a composite function x ∈ Rd ↦ G(f(x), x). Assuming sufficient smoothness for G, we give a “tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f, with a coefficient depending on G and on the Ha norm of f, for all integers n, a, d with a > d/2. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the Hn norm of the function x ↦ G(f(x), x). When applied to the case G(f(x), x) = f2(x), this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.

Quantum Harmonic Analysis and Geometric Invariants

We prove results on complex interpolation of vector‐valued Sobolev spaces over the half‐line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space‐valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.

We consider the imbedding inequalit), I1.(). sr,,,dll IIn,fnu; H (R) is the Sobolev space (or Bessel potential space) ofL type and (integer or fa6tional) order n. We write down upper bounds for the constants Sr,n,a, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if (r= 2 or) n > d/2, r= oe, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on Sr,n,d for n > d/2, 2 < r < oo; in many cases these are close to the previous upper bounds, as illustrated by a number ofexamples, thus characterizing the sharp constants with little uncertainty.

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products onℝ+). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spacesLωpℝ+forp≥1. We also show new inequalities in these weighted Lebesgue spaces. These results are applied to several concrete function spaces, for example, weighted Sobolev spaces and fractional Sobolev spaces defined by Weyl fractional derivation.

Let denote the spaces of Bessel potentials as denned and discussed in [6]. When 1 &lt; q &lt; ∞ and α is an integer , the Sobolev space which consists of functions Fin Lq with α derivatives in Lq and with norm If now we make the change of variables x = ev, f(x) = F (In x) it is easily seen that the ratio

The Cauchy problem for the nonlinear Schrodinger equation is called unconditionally well posed in a data space \(E\) if it is well posed in the usual sense and the solution is unique in the space \(C([0,T]; E)\). In this paper, this notion of the unconditional well-posedness is redefined so that it covers \(L^p\)-based Sobolev spaces as data space \(E\) and it is equivalent to the usual one when \(E\) is an \(L^2\)-based Sobolev space \(H^s\) Based on this definition, it is shown that the Cauchy problem for the 1D cubic NLS is unconditionally well posed in Bessel potential spaces \(H^s_p\) for \(4/3&lt;p\le 2\) under certain assumptions on \(s\). For more information see https://ejde.math.txstate.edu/Volumes/2025/41/abstr.html

In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of p- Laplacian type. The problem in its variational form is as follows: \displaystyle \text{min} \left\{ \int\limits_{\Omega \cap \{v&gt;0\}} \left(\frac{1}{p}|\nabla v|^p + \lambda_{+}^p+ f_{+}v \right)dx + \int\limits_{\Omega \cap \{v\leq 0\}} \left(\frac{1}{q}|\nabla v|^q + \lambda_{-}^q+ f_{-}v\right)dx \right\}. Here we minimize among all admissible functions v in an appropriate Sobolev space with a prescribed boundary datum v=g on \partial \Omega . First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where p and q go to infinity, obtaining a limiting free boundary problem governed by the \infty -Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.

Let = ( ( )) be the abstract space of Bessel potentials and a positive smooth Radon measure on . For 2 , we give necessary and sufficient criteria for the boundedness of from ( ) into ( ), provided is contractive. Among others, we shall prove that the boundedness is equivalent to a capacitary type inequality. Further we give necessary and sufficient conditions for to be compactly embedded in ( ). Our method relies essentially on establishing a capacitary strong type inequality.

Sobolev spaces and their embedding properties have long been of central importance in the study of partial differential equations, particularly for the classical theoretical analysis of both linear and nonlinear problems. In recent years, computer‐assisted proof techniques have been developed for obtaining existence and uniqueness proofs of solutions to a variety of nonlinear partial differential equations that arise in applications, and they have led to a number of new results that currently lie beyond the reach of classical analytical approaches. The use of computer‐assisted methods, however, frequently relies on the explicit knowledge of a variety of embedding constants for Sobolev spaces. In the present paper, we show that in the context of certain Sobolev space Banach algebras, these constants themselves can be bounded rigorously and precisely using validated computations based on interval arithmetic, combined with analytical error estimates.

Riccati map on [formula omitted] and its applications

‎I‎. ‎Kaplansky showed in 1947 that every submultiplicative norm $\Vert‎ . ‎\Vert$ on the algebra ${\mathcal C}(K)$ of complex--valued‎ ‎functions on an infinite compact space $K$ satisfies $\Vert f \Vert‎ ‎\ge \Vert f \Vert _K$ for every $f \in {\mathcal C}(K),$ where‎ ‎$\Vert f \Vert _K=max_{t \in K} \vert f(t)\vert $ denotes the‎ ‎standard norm on ${\mathcal C}(K).$ He asked whether all‎ ‎submultiplicative norms $\Vert‎ . ‎\Vert$ were in fact equivalent to‎ ‎the standard norm (which is obviously true for finite compact‎ ‎spaces)‎, ‎or equivalently‎, ‎whether all homomorphisms from ${\mathcal‎ ‎C}(K)$ into a Banach algebra were continuous‎. ‎This problem turned‎ ‎out to be undecidable in ZFC‎, ‎and we will discuss here some recent‎ ‎progress due to Pham and open questions concerning the structure of‎ ‎the set of nonmaximal prime ideals of ${\mathcal C}(K)$ which are‎ ‎closed with respect to a discontinuous submultiplicative norm on‎ ‎${\mathcal C}(K)$ when the continuum hypothesis is assumed‎. ‎We will‎ ‎also discuss the existence of discontinuous characters on Fr\'echet‎ ‎algebras (Michael's problem)‎, ‎a long standing problem which remains‎ ‎unsolved‎. ‎The Mittag--Leffler theorem on inverse limits of complete‎ ‎metric spaces plays an essential role in the literature concerning‎ ‎both problems‎.

In the present paper we collect results on Mellin convolution equations (MCEs), obtained recently. We start with the motivation and in the first section are exposed MCEs which encounter in applications. Further we expose the boundedness results of corresponding operators in the Lebesgue space with weight. Fourier convolution (Wiener-Hopf) equations are defined and their connection to MCEs is described. Solvability and Fredholm properties and the index formulae for MCEs are formulated in terms of the symbol functions assigned to them. Results on the Banach algebra generated by Mellin and Wiener-Hopf operators in the Lebesgue space are exposed: The symbol function is defined and the Fredholm criteria is formulated, the index formula is written. In conclusion we expose relatively new results on Fredholm property and index of MCEs in the Bessel potential spaces. These results are applied to the mixed boundary value problem (Mixed BVP) for the Laplace equation in an angle, which reduces to an equivalent MCE. Results on the Fredholm property, solvability and index of such equations are formulated and, in conclusion, applied to the above formulated mixed BVP for the Laplace equation.

In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential operators criteria in terms of kernels may be less effective (Carleman's example), our criteria are given in terms of the operators' symbols defined on the noncommutative analogue of the phase space $G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and $\hat{G}$ is its unitary dual. We also show results concerning general non-invariant operators as well as Schatten properties on Sobolev spaces. A trace formula is derived for operators in the Schatten class $S_{1}(L^{2}(G))$. Examples are given for Bessel potentials associated to sub-Laplacians (sums of squares) on compact Lie groups, as well as for powers of the sub-Laplacian and for other non-elliptic operators on SU(2)$\simeq\mathbb S^3$ and on SO(3).