Volume 36, Issue 5

In this note, we characterize the essential numerical range of a block diagonal operator T = ⊕ i T i {T=\bigoplus_{i}T_{i}} in terms of the numerical ranges { W ⁢ ( T i ) } i {\{W(T_{i})\}_{i}} of its components. Specifically, the essential numerical range of T is the convex hull of the limit superior of { W ⁢ ( T i ) } i {\{W(T_{i})\}_{i}} . This characterization can be simplified further. In fact, we prove the existence of a decomposition of T for which the convex hull is not required.

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of a 3 ⁢ b {a^{3}b} -quadrilaterals with some irrational angle: there are a sequence of 1-parameter families of quadrilaterals admitting 2-layer earth map tilings together with their basic flip modifications under extra condition, and 5 sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end.

We prove Schauder estimates for elliptic and parabolic problems governed by the degenerate operator ℒ = Δ x + D y ⁢ y + c y ⁢ D y , \mathcal{L}=\Delta_{x}+D_{yy}+\frac{c}{y}D_{y}, in the half-space Ω = { ( x , y ) : x ∈ ℝ N , y > 0 } {\Omega=\{(x,y):x\in\mathbb{R}^{N},y>0\}} , under Neumann boundary conditions at y = 0 {y=0} .

We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1 {p<1} . As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand’s results.

In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional p -Laplace problem: { ( - Δ ) p s ⁢ v + 𝒱 ⁢ ( ξ ⁢ x ) ⁢ | v | p - 2 ⁢ v = λ ⁢ | v | p - 2 ⁢ v + f ⁢ ( v )   in  ⁢ ℝ N , ∫ ℝ N | v | p ⁢ 𝑑 x = a p , \left\{\begin{aligned} \displaystyle{}(-\Delta)_{p}^{s}v+\mathcal{V}(\xi x)% \lvert v\rvert^{p-2}v&\displaystyle=\lambda\lvert v\rvert^{p-2}v+f(v)\quad% \text{in }\mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}}\lvert v\rvert^{p}\,dx&\displaystyle=a^{p},% \end{aligned}\right. where a , ξ > 0 {a,\xi>0} , p ≥ 2 {p\geq 2} , λ ∈ ℝ {\lambda\in\mathbb{R}} is an unknown parameter that appears as a Lagrange multiplier, 𝒱 : ℝ N → [ 0 , ∞ ) {\mathcal{V}:\mathbb{R}^{N}\to[0,\infty)} is a continuous function, and f is a continuous function with L p {L^{p}} -subcritical growth. We prove that there exists the multiplicity of solutions by using the Lusternik–Schnirelmann category. Next, we show that the number of normalized solutions is at least the number of global minimum points of 𝒱 {\mathcal{V}} , as ξ is small enough via Ekeland’s variational principle.

We define L -series of weakly holomorphic quasimodular forms and we derive functional equations of those L -series. We also prove a converse theorem for weakly holomorphic quasimodular forms.

Using a F 6 7 {{}_{7}F_{6}} hypergeometric transformation formula, we prove two supercongruences. In particular, one of these supercongruences confirms a recent conjecture of Guo, Liu and Schlosser, and gives an extension of a supercongruence of Long and Ramakrishna.

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.

We show that the modular isomorphism problem has a positive answer for groups of nilpotency class 2 with cyclic center, i.e., that for such p -groups G and H an isomorphism between the group algebras FG and FH implies an isomorphism of the groups G and H for F the field of p elements. For groups of odd order this implication is also proven for F being any field of characteristic p . For groups of even order we need either to make an additional assumption on the groups or on the field.

Let 𝔽 q d {\mathbb{F}_{q}^{d}} be the d -dimensional vector space over the finite field 𝔽 q {\mathbb{F}_{q}} with q elements. For each non-zero r in 𝔽 q {\mathbb{F}_{q}} and E ⊂ 𝔽 q d {E\subset\mathbb{F}_{q}^{d}} , we define W ⁢ ( r ) {W(r)} as the number of quadruples ( x , y , z , w ) ∈ E 4 {(x,y,z,w)\in E^{4}} such that Q ⁢ ( x - y ) Q ⁢ ( z - w ) = r {\frac{Q(x-y)}{Q(z-w)}=r} , where Q is a non-degenerate quadratic form in d variables over 𝔽 q {\mathbb{F}_{q}} . When Q ⁢ ( α ) = ∑ i = 1 d α i 2 {Q(\alpha)=\sum_{i=1}^{d}\alpha_{i}^{2}} with α = ( α 1 , … , α d ) ∈ 𝔽 q d {\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{F}_{q}^{d}} , Pham (2022) recently used the machinery of group actions and proved that if E ⊂ 𝔽 q 2 {E\subset\mathbb{F}_{q}^{2}} with q ≡ 3 ⁢ ( mod ⁡  4 ) {q\equiv 3~{}(\operatorname{mod}\,4)} and | E | ≥ C ⁢ q {|E|\geq Cq} , then we have W ⁢ ( r ) ≥ c ⁢ | E | 4 q {W(r)\geq\frac{c|E|^{4}}{q}} for any non-zero square number r ∈ 𝔽 q {r\in\mathbb{F}_{q}} , where C is a sufficiently large constant, c is some number between 0 and 1, and | E | {|E|} denotes the cardinality of the set E . In this article, we improve and extend Pham’s result in two dimensions to arbitrary dimensions with general non-degenerate quadratic distances. As a corollary of our results, we also generalize the sharp results on the Falconer-type problem for the quotient set of distance set due to the first two authors and Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. The key new ingredient is to relate the estimate of the W ⁢ ( r ) {W(r)} to a quadratic homogeneous variety in 2 ⁢ d {2d} -dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums which are more handleable than the Kloosterman sums appearing in the standard distance-type problems.

We consider a very global q -integral transform, essentially characterized by having a bounded kernel and satisfying a set of natural and useful properties for the realization of applications. The main ambition of this work is to seek conditions that guarantee uncertainty principles of the Donoho–Stark type for that class of q -integral transforms. It should be noted that the global character of the q -integral transform in question allows one to immediately deduce corresponding Donoho–Stark uncertainty principles for q -integral operators that are its particular cases. These particular cases are very well-known operators, namely: a q -cosine-Fourier transform, a q -sine-Fourier transform, a q -Fourier transform, a q -Bessel–Fourier transform and a q -Dunkl transform. Moreover, generalizations of the local uncertainty principle of Price for the q -cosine-Fourier transform, q -sine-Fourier transform, q -Fourier transform, q -Bessel–Fourier transform and q -Dunkl transform are also obtained.

We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at p = ∞ {p=\infty} , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , ( λ + 1 ) 2 ) {[\lambda^{2},(\lambda+1)^{2})} to [ λ 2 , ( λ + ϵ ⁢ ( λ ) ) 2 ) {[\lambda^{2},(\lambda+\epsilon(\lambda))^{2})} , where ϵ ⁢ ( λ ) {\epsilon(\lambda)} is a function of λ that goes to 0 as λ goes to ∞ {\infty} . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p L^{p} -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527].

In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely 𝐒𝐆 {\mathbf{SG}} -class.

Distinctive characteristics of Iwanaga–Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.

This corrigendum corrects Proposition 5.2 in [A. Dasgupta, L. Mohan and S. S. Mondal, Multilinear Fourier Integral operators on modulation spaces, Forum Math. 2024, 10.1515/forum-2023-0158].