论文标题
hardy-小木 - $ p = 1 $的不平等现象
Hardy--Littlewood--Sobolev inequality for $p=1$
论文作者
论文摘要
令$ \ mathcal {w} $为$ \ mathbb {r}^\ ell $ - valued schwartz分布的$ \ mathbb {r}^\ ell $ valued schwartz分布的封闭式扩张和不变子空间。我们表明，如果空间$ \ MATHCAL {W} $不包含类型$ a \ otimesΔ_0$，$Δ_0$作为Dirac Delta的分布，则不等式$ \ | \ | \ | \ Mathbb {i}_α[i}_α[f] f] f] \ | ___ {p，1 {p，1} p，1} \ | f \ | _ {l_1} $，$ \ frac {p-1} {p} = \fracα{d} $，对于函数$ f \ in \ mathcal {w} \ cap l_1 $，均具有均匀的常数；这里$ \ mathbb {i}_α$是订单$α$的潜力，$ l_ {p，1} $是lorentz空间。该结果意味着特殊情况，不等式$ \ | \ nabla^{m-1} f \ | _ {l _ {\ frac {\ frac {d} {d-1}，1}}} \ lyseSim \ | a f \ | a f \ | _ f \ | _ {l_1} $ a $ a $ a $ a $ a $ cAUTCAMIAL ELLIPT DROCE $ ellipt Ellipt elipt nipt uptial nipt nipt uptial nipt coplipt $ $ $ m。
Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type $a\otimes δ_0$, $δ_0$ being the Dirac delta, then the inequality $\|\mathbb{I}_α[f]\|_{L_{p,1}}\lesssim \|f\|_{L_1}$, $\frac{p-1}{p} = \fracα{d}$, holds true for functions $f\in\mathcal{W}\cap L_1$ with a uniform constant; here $\mathbb{I}_α$ is the Riesz potential of order $α$ and $L_{p,1}$ is the Lorentz space. This result implies as a particular case the inequality $\|\nabla^{m-1} f\|_{L_{\frac{d}{d-1},1}} \lesssim \|A f\|_{L_1}$, where $A$ is a canceling elliptic differential operator of order $m$.
