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Poincaré-Sobolev and isoperimetric inequalities, maximal functions, and half-space estimates for the gradient. (English) Zbl 0831.46032

Krbec, Miroslav (ed.) et al., Nonlinear analysis, function spaces and applications. Vol. 5. Proceedings of the spring school held in Prague, May 23-28, 1994. Prague: Prometheus Publishing House. 231-265 (1994).
This is a well written survey consisting of 4 lectures. The first lecture collects more or less known inequalities in weighted \(L_p\)-spaces for the fractional maximal operator \[ M_\alpha f(x)= \sup_{x\in B} |B|^{{\alpha\over n}- 1} \int_B |f(y)|dy\qquad (B:\text{ ball}) \] in \(\mathbb{R}^n\) and in homogeneous spaces. The second lecture concentrates on similar inequalities of Poincaré type \[ \Biggl(-\hskip-.9em\int_B |f(z)- f_B|^q u(z)dz\Biggr)^{{1\over q}}\leq cr\Biggl(-\hskip-.9em\int_B \sum_j |X_j f(z)|^p v(z)dz\Biggr)^{{1\over p}}, \] where \(X_j\) are first-order vector fields, \(r\) is the radius of \(B\). The main bulk of the paper consists of lecture 3 dealing with recent results for half- space estimates for the gradient, \[ \Biggl( \int_{\mathbb{R}^{n+ 1}_+} |\nabla f(x, y)|^q w(x, y) dx dy\Biggr)^{{1\over q}}\leq c\Biggl( \int_{\mathbb{R}^n} |f(x)|^p v(x)dx\Biggr)^{{1\over p}}, \] where \(f(x, y)= f* k_y(x)\), \(k_y(x)= y^{- n} k\left({x\over y}\right)\) is a dilated smooth kernel with some decay at infinity. The final lecture 4 deals with inequalities for generalized operators of Riesz type of the form \(I_{\alpha, \beta} f(x)= f* h_{\alpha, \beta}(x)\) with \(h_{\alpha, \beta}(x)= |x|^{1+ \alpha- n}|x_n|^{\beta- 1}\).
For the entire collection see [Zbl 0811.00017].
Reviewer: H.Triebel (Jena)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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