Vanishing exponential integrability for functions whose gradients belong to \(L^n(\log(e+L))^{\alpha}\)
Recently, there has been considerable interest in extending the classical Trudinger exponential-type inequality to other situations. One such extension appears in the paper by \textit{D. E. Edmunds} and \textit{R. Hurri-Svrjanen} [Isr. J. Math. 123, 61-92 (2001; Zbl 0991.46019)] showing that if \(u = u(x) \) is a weakly differentiable function on \(\mathbb R^n\) which satisfies \[ \int_B \Phi_{\alpha}(\nabla u) da \leq 1, \] where \(\Phi_{\alpha}(t) = t^n(\log(e + t))^\alpha \), \(t > 0\) and \(0 \leq \alpha \leq n -1\), then there are positive constants \(a\) and \(c_{\alpha}\) independent of \(u\) such that \[ \oint_B \exp (a u(x) - u_B ^{n/(n-1-\alpha)})dx \leq c_{\alpha} \tag{1.1} \] for any ball \(B \subset \mathbb R^n\) when \(\alpha < n-1\). When \(\alpha = n -1\), there is an additional constant \(b > 0\) independent of \(u\) such that \[ \oint_B \exp(a\exp(b u(x) - u_B ^{n/(n-1)}))dx \leq c_{n-1}. \tag{1.2} \] One problem with estimates (1.1) and (1.2) is that they give very little information as to the behavior of these averages when the radius of the ball \(B\) tends to zero. In the paper under review, the authors ask whether is it possible to refine these estimates so that the constants \(c_{\alpha}\) are independent of both \(u\) and \(B\). When \(\alpha = 0\), it is proven in Proposition 3.3 of the first author's notes [Lectures on \(L^p\)-Potential Theory, University of Umea] that \[ \oint_{B(x_0,r)} \left[ \exp(a u(x) - u(x_0) ^{p/(p-1)}) - 1 \right]dx = o(1) \tag{1.3} \] as \(r\) tends to zero, for almost every \(x_0\), modulus \(C_{\beta,p}\), the Bessel capacity, \(\beta p = n\), some constant \(a> 0\) independent of \(u\) and \(r\), with \(u = G_{\beta} \ast f\), a Bessel potential of order \(\beta > 0\), and \(f\) an \(L^p\)-integrable function on \(R^n\). In particular, when \(\beta = 1\), \[ \oint_{B(x_0,r)} \left[ \exp(a u(x) - u(x_0) ^{n/(n-1)}) - 1 \right]dx = o(1) \tag{1.4} \] \(C_{1,n}\)- a.e. \(x_0\) when \(\int \bigtriangledown u ^n dx < \infty\). On the other hand, for \(\Phi_{\alpha}\), a result by \textit{A. Cianchi} and \textit{B. Stroffolini} [J. Math. Anal. Appl. 227, 166-186 (2001; Zbl 0914.42014)] (1.4) only implies \[ \oint_{B(x_0,r)} \left[ \exp(a \exp(b u(x) - u(x_0))^{n/(n-1) - \delta}) - e^a \right]dx = o(1) \tag{1.5} \] as \(r\) tends to zero, for every \(\delta >0\) and \(C_{1,\Phi_{n-1}}\) - a.e. \(x_0\) and some constant \(a > 0\) independent of \(u\). It is unknown if (1.5) holds for \(\delta = 0\). The main result of the present paper (Theorem 1.6) is a subsitute for (1.5) when \(\delta = 0\) which is reproduced below. Theorem. Let \(\Psi_{\alpha}(t) = t^n (\log (e + \frac{1}{t}))^{- \alpha}\), \(t > 0\) and \(0 \leq \alpha n - 1\), then \(\int \Psi_{\alpha}(\nabla u)dx < \infty\) implies \[ \oint _{B(x_0,r)} \left[\exp \left (a \Psi_{\alpha}( u(x) - u(x_0))^{\frac{1}{n-1- \alpha}} \right) - 1 \right]dx = o(1) \] \(C_{\Psi_{\alpha}}\)- a.e. \(x_0\), \(\alpha < n - 1\), as \(r \to 0\), and \[ \oint _{B(x_0,r)} \left[\exp \left (a (b\Psi_{n-1}( u(x) - u(x_0))^{\frac{1}{n-1}} \right) - e^a1 \right]dx = o(1) \] \(C_{\Psi_{n-1)}}\)- a.e. \(x_0\), as \(r \to 0\), when \(\alpha = n - 1\), for some constants \(a > 0\) and \(b > 0\). Capacities \(C_{\Psi_{\alpha}}\) are introduced and their basic properties are established.
scientific article; zbMATH DE number 1895819