We study the derivative operator of the generalized spherical mean S
 γ
 . By considering a more general multiplier $$\mathfrak{m}_{\gamma,b}^\Omega\;=\;V_{\frac{n-2}{2}+\gamma}(\mid\xi\mid)\mid\xi\mid^b\Omega(\xi\prime)$$
 and finding the smallest γ such that $$\mathfrak{m}_{\gamma,b}^\Omega$$
 is an Hp multiplier, we obtain the optimal range of exponents (γ, β, p) to ensure the Hp(ℝn) boundedness of ∂βS
 1
 γ
 f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on Hp(ℝn) spaces.

Derivative estimates of averaging operators and extension

In this paper, we study the K-moduli spaces of pairs ( X, c (cid:80) L i ), where X is a del Pezzo surface of degree d , L i ’s are the lines on it counted with multiplicity, and c > 0 any coeﬃcient such that this pair is log Fano. We prove that for d = 2 , 3 , 4, there are no walls for the K-moduli spaces. For d = 1, we exhibit examples of walls.

pdf/k-moduli-spaces-of-del-pezzo-pairs-with-sum-of-lines-and-2jc7kv1l.pdf

K-Moduli Spaces of Del Pezzo Pairs with Sum of Lines and Wall Crossings

In the paper, we study derivative estimates of the iterated spherical averages (At)N(f). We obtain the optimal range of exponents (α, N, p) to ensure the Lp boundedness of $$P\left( {{\partial \over {\partial x}}} \right){\left( {{A_1}} \right)^N}(f)$$
 for 1 ≤ p ≤ ∞, where P is a homogeneous polynomial of degree α. The main theorem extends some known results. As an application, we obtain the smallest N such that (A1)N: Lp(ℝn) → L
 (ℝn).

Derivative Estimates of Iterated Spherical Average and Extension

The moduli space of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth‐century work of Cayley and Salmon. Modern interest in was restored in the 1980s by Naruki's explicit construction of a ‐equivariant smooth projective compactification of , and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd‐Barron–Alexeev (KSBA) stable pair compactification of as a natural sequence of blowups of . We describe generators for the cones of ‐invariant effective divisors and curves of both and . For Naruki's compactification , we further obtain a complete stable base locus decomposition of the ‐invariant effective cone, and as a consequence find several new ‐equivariant birational models of . Furthermore, we fully describe the log minimal model program for the KSBA compactification , with respect to the divisor , where is the boundary and is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.

The W(E6)$W(E_6)$‐invariant birational geometry of the moduli space of marked cubic surfaces

A general curve $C$ of genus six is canonically embedded into the smooth del Pezzo surface $\Sigma \subseteq \mathbb{P}^1 \times \mathbb{P}^2$ of degree $5$ as a divisor in the class $\mathcal{O}_{\Sigma}(2,2)$. In this article, we study the variation of geometric invariant theory (VGIT) for such pairs $(\Sigma,C)$, and relate the VGIT moduli spaces to the K-moduli of pairs $(\Sigma,C)$ and the Hassett-Keel program for moduli of genus six curves. We prove that the K-moduli spaces ${\overline{M}}^{K}(c)$ give the final several steps in the Hassett-Keel program for ${\overline{M}}_6$.

The Moduli Space of Genus Six Curves and K-stability: VGIT and the Hassett-Keel Program

In the paper, we study some multiplier operator $$\mu _{\gamma ,\alpha }$$ raised from studying the $$L^{p}$$-approximation of the spherical mean $$S_{t}^{\gamma }.$$ We obtain the optimal range of exponents $$ (\alpha ,\gamma ,p)$$ such that $$\mu _{\gamma ,\alpha }$$ is an $$L^{p}$$ multiplier. As an application, we obtain the convergence rate for $$ S_{t}^{\gamma }\left( f\right) \ $$in the $$L^{p}$$ spaces.

Certain averaging operators on Lebesgue spaces

Let X be the blow-up of P 2 along m general points, and A = H − P ε i E i be a generic polarization with 0 < ε i ≪ 1. We classify the Chern characters which satisfy the weak Brill-Noether property, i.e. a general sheaf in M A ( v ), the moduli space of slope stable sheaves with Chern character v , has at most one non-zero cohomology. We further give a necessary and suﬃcient condition for the existence of stable sheaves. Our strategy is to specialize to the case when the m points are collinear.

Moduli Spaces of Sheaves on General Blow-ups of $\mathbb{P}^2$

Abstract In this article, we study the boundedness properties of the averaging operator S t γ on Triebel-Lizorkin spaces $$\dot F_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mover><mml:mi>F</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>ℝ</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> for various p,q . As an application, we obtain the norm convergence rate for S t γ ( f ) on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of S t γ is given. 

/pdf/certain-averaging-operators-on-triebel-lizorkin-spaces-2imq632j.pdf

Certain averaging operators on Triebel-Lizorkin spaces

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