Dilation-balanced product Pitt inequalities for mixed-tail potentials

Abstract

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 \[ \|\Pi_{\alpha}\widehat f_{A}\|_{L_{\vec q}(\mathbb R^d)} \le C\|\Pi_{\beta}f\|_{L_{\vec p}(\mathbb R^d)}, \qquad \beta_k=1-\frac1{p_k}-\frac1{q_k}-\alpha_k, \] where $1\le p_k\le q_k<\infty$ and $\alpha_k>-1/q_k$. 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 $-1/q_k<\alpha_k<1-1/q_k$ is exactly the range in which both weighted norms are finite.