Analysis and Mathematical Physics最新文献

Let (kge 1), (0le alpha <d) and (mathfrak {M}_{b,alpha }^k) be the k-th order fractional maximal commutator. When (alpha =0), we denote (mathfrak {M}_{b,alpha }^k=mathfrak {M}_{b}^k). We show that (mathfrak {M}_{b,alpha }^k) is bounded from the first order Sobolev space (W^{1,p_1}(mathbb {R}^d)) to (W^{1,p}(mathbb {R}^d)) where (1<p_1,p_2,p<infty ), (1/p=1/p_1+k/p_2-alpha /d). We also prove that if (0<s<1), (1<p_1,p_2,p,q<infty ) and (1/p=1/p_1+k/p_2), then (mathfrak {M}_b^k) is bounded and continuous from the fractional Sobolev space (W^{s,p_1}(mathbb {R}^d)) to ({W^{s,p}(mathbb {R}^d)}) if (bin W^{s,p_2}(mathbb {R}^d)), from the inhomogeneous Triebel–Lizorkin space (F_s^{p_1,q}(mathbb {R}^d)) to (F_s^{p,q}(mathbb {R}^d)) if (bin F_s^{p_2,q} (mathbb {R}^d)) and from the inhomogeneous Besov space (B_s^{p_1,q}(mathbb {R}^d)) to (B_s^{p,q}(mathbb {R}^d)) if (bin B_s^{p_2,q}(mathbb {R}^d)). It should be pointed out that the main ingredients of proving the above results are some refined and complex difference estimates of higher order maximal commutators as well as some characterizations of the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces.

In this paper, we focus on investigating the entire solutions to one certain type of non-linear binomial differential equations with respect to several problems posed by Gundersen and Yang. We also illustrate the exponential polynomials solutions to this equation. Some examples are used to illustrate our results.

In this paper, we consider the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. First, we show the uniqueness for recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we address the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. Furthermore, we examine the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable with respect to noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance by the direct sampling method.

We present a comprehensive class of Sobolev bi-orthogonal polynomial sequences, which emerge from a moment matrix with an LU factorization. These sequences are associated with a measure matrix defining the Sobolev bilinear form. Additionally, we develop a theory of deformations for Sobolev bilinear forms, focusing on polynomial deformations of the measure matrix. Notably, we introduce the concepts of Christoffel–Sobolev and Geronimus–Sobolev transformations. The connection formulas between these newly introduced polynomial sequences and existing ones are explicitly determined.

We consider the weighted Bergman space (A^2_psi ) of all holomorphic functions on ({textbf{B}_n}) square integrable with respect to an exponential weight measure (e^{-{psi }} dV) on ({textbf{B}_n}), where

We characterize boundedness (or compactness) of Toeplitz operators and Hankel operators on (A^2_psi ).

In this paper, we study the existence of positive solutions for a class of Kirchhoff equation with critical growth

where (a>0), (b>0), (Vin L^frac{3}{2}(Omega )) is a given nonnegative function and (Omega subseteq mathbb {R}^3) is an exterior domain, that is, an unbounded domain with smooth boundary (partial Omega ne emptyset ) such that (mathbb {R}^3backslash Omega ) non-empty and bounded. By using barycentric functions and Brouwer degree theory to prove that there exists a positive solution (uin D^{1,2}_0(Omega )) if (mathbb {R}^3backslash Omega ) is contained in a small ball.

We consider a class of hypoelliptic operators of the following type

where ((a_{ij})), ((b_{ij})) are constant matrices and ((a_{ij})) is symmetric positive definite on ({mathbb {R}}^{p_0}) ((p_0le N)). We obtain generalized Hölder estimates for ({mathcal {L}}) on ({mathbb {R}}^{N+1}) by establishing several estimates of singular integrals in generalized Morrey spaces.

For G an open set in ({mathbb {C}}) and W a non-vanishing holomorphic function in G, in the late 1990’s, Pritsker and Varga (Constr Approx 14, 475-492 1998) characterized pairs (G, W) having the property that any f holomorphic in G can be locally uniformly approximated in G by weighted holomorphic polynomials ({W(z)^np_n(z)}, deg(p_n)le n). We further develop their theory in first proving a quantitative Bernstein-Walsh type theorem for certain pairs (G, W). Then we consider the special case where (W(z)=1/(1+z)) and G is a loop of the lemniscate ({zin {mathbb {C}}: |z(z+1)|=1/4}). We show the normalized measures associated to the zeros of the (n-th) order Taylor polynomial about 0 of the function ((1+z)^{-n}) converge to the weighted equilibrium measure of ({overline{G}}) with weight |W| as (nrightarrow infty ). This mimics the motivational case of Pritsker and Varga (Trans Amer Math Soc 349, 4085-4105 1997) where G is the inside of the Szegő curve and (W(z)=e^{-z}). Lastly, we initiate a study of weighted holomorphic polynomial approximation in ({mathbb {C}}^n, n>1).

This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that the only obstruction for having norm resolvent convergence is the existence of an eigenvalue of infinite multiplicity for the limiting operator. More precisely, we prove the convergence of the resolvents in operator norm once projected into the orthogonal of the corresponding eigenspace.

We are interested in four-dimensional Dirac–Klein–Gordon equations, a fundamental model in particle physics. The main goal of this paper is to establish global existence of solutions to the coupled system and to explore their long-time behavior. The results are valid uniformly for mass parameters varying in the interval [0, 1].