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1. On the Strict Majorant Property in Arbitrary Dimensions

   https://doi.org/10.1093/qmath/haac021  

   Gressman, P T; Guo, S; Pierce, L B; Roos, J; Yung, P -L (July 2022, The Quarterly Journal of Mathematics)

   Abstract In this work we study d-dimensional majorant properties. We prove that a set of frequencies in $$\mathbb{Z}^d$$ satisfies the strict majorant property on $L^p([0,1]^d)$ for all p > 0 if and only if the set is affinely independent. We further construct three types of violations of the strict majorant property. Any set of at least d + 2 frequencies in $$\mathbb{Z}^d$$ violates the strict majorant property on $L^p([0,1]^d)$ for an open interval of $$p \not\in 2\mathbb{N}$$ of length 2. Any infinite set of frequencies in $$\mathbb{Z}^d$$ violates the strict majorant property on $L^p([0,1]^d)$ for an infinite sequence of open intervals of $$p \not\in 2\mathbb{N}$$ of length 2. Finally, given any p > 0 with $$p \not\in 2\mathbb{N}$$, we exhibit a set of d + 2 frequencies on the moment curve in $$\mathbb{R}^d$$ that violate the strict majorant property on $L^p([0,1]^d).$ 

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2. Marcinkiewicz Averages of Smooth Orthogonal Projections on Sphere

   https://doi.org/10.1007/s00041-022-09966-y  

   Bownik, Marcin; Dziedziul, Karol; Kamont, Anna (October 2022, Journal of Fourier Analysis and Applications)

   Abstract We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $$\Gamma \subset \mathbb {R}^d$$ Γ ⊂ R d , a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $$\mathbb {R}^d/\Gamma $$ R d / Γ . We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of SO ( d ), is a multiple of the identity on $$L^2(\mathbb {S}^{d-1})$$ L 2 ( S d - 1 ) . As an application we construct highly localized continuous Parseval frames on the sphere. 

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3. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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4. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

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5. Asymptotics of singular values for quantum derivatives

   https://doi.org/10.1090/tran/8827  

   Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy (March 2023, Transactions of the American Mathematical Society)

   We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f f from the homgeneous Sobolev space W ˙ d 1 ( R d ) \dot {W}^1_d(\mathbb {R}^d) on R d . \mathbb {R}^d. The asymptotic coefficient ‖ ∇ f ‖ L d ( R d ) \|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal L d , ∞ , \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C ∗ C^{\ast } -algebraic notion of the principal symbol mapping on R d \mathbb {R}^d , as developed recently by the last two authors and collaborators. 

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