AI 中文总结
受一般单调函数理论启发，为乘积傅里叶不等式引入多参数一般单调性类，用混合尾条件取代单调性条件，证明了各向异性混合范数估计，该类包含多种函数，给出特定紧支集例子中加权范数有限的范围。
AI 中文摘要
受一维和径向一般单调函数理论的启发，我们为乘积傅里叶不等式引入了一个自然的多参数一般单调性类。单调性条件被一个内在的混合尾条件所取代：正象限上的函数是复 Radon 测度的上尾势，且该测度的积分全变差由函数的局部积分控制。通过全混合分布导数恢复表示测度，所以该条件是内在的而非逐坐标的。对于逐坐标的偶延拓，我们证明了各向异性混合范数估计\[ \|\Pi_{\alpha}\widehat f_{A}\|_{L_{\vec q}(\mathbb R^d)} \le C\|\Pi_{\beta}f\|_{L_{\vec p}(\mathbb R^d)}, \qquad \beta_k=1-\frac1{p_k}-\frac1{q_k}-\alpha_k, \]其中$1\le p_k\le q_k\lt\infty$且$\alpha_k\gt -1/q_k$。这里$\widehat f_A$是阿贝尔求和傅里叶变换，对于$f\in L^1$，它几乎处处与普通傅里叶变换一致。指数关系由独立的坐标扩张强制产生。因此，该定理具有与各向异性单调理论相同的逐坐标扩张平衡，而假设是按照一般单调性的精神制定的：变差由局部控制，且不施加乘积或逐坐标单调性结构。该类包含所有扇形混合尾势、由满足混合矩条件的任意非乘积正测度生成的无限维锥，以及在坐标变量中不单调的实非可分函数。对于在原点邻域内由正的常数下界界定的紧支集例子，范围$-1/q_k\lt\alpha_k\lt1 - 1/q_k$恰好是两个加权范数都有限的范围。
英文摘要
Motivated by the one-dimensional and radial theory of general monotone functions, we introduce a natural multiparameter general-monotonicity class for product Fourier inequalities. The monotonicity condition is replaced by an intrinsic mixed-tail condition: a function on the positive orthant is the upper-tail potential of a complex Radon measure, and the integrated total variation of this measure is controlled by a local integral of the function. The representing measure is recovered as the full mixed distributional derivative, so the condition is intrinsic rather than coordinatewise. For coordinatewise even extensions we prove the anisotropic mixed-norm estimate \[ \|Π_α\widehat f_{A}\|_{L_{\vec q}(\mathbb R^d)} \le C\|Π_βf\|_{L_{\vec p}(\mathbb R^d)}, \qquad β_k=1-\frac1{p_k}-\frac1{q_k}-α_k, \] where $1\le p_k\le q_k<\infty$ and $α_k>-1/q_k$. Here $\widehat f_A$ is an Abel-summed Fourier transform and agrees almost everywhere with the ordinary Fourier transform for $f\in L^1$. The exponent relation is forced by independent coordinate dilations. Thus the theorem has the same coordinatewise dilation balance as the anisotropic monotone theory, while the hypotheses are formulated in the spirit of general monotonicity: variation is controlled locally, and no product or coordinatewise monotonicity structure is imposed. The class contains all sectorial mixed-tail potentials, an infinite-dimensional cone generated by arbitrary non-product positive measures satisfying the mixed-moment condition, and real nonseparable functions which are not monotone in the coordinate variables. For compactly supported examples which are bounded below by a positive constant in a neighborhood of the origin, the range $-1/q_k<α_k<1-1/q_k$ is exactly the range in which both weighted norms are finite.
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