%%%% TYPESET WITH PLAIN TEX. NEEDS AMS - FONTS. \magnification=\magstep1 \baselineskip=22pt \input amssym.def %%%%% AMS - FONTS !! \font\eightit=cmti8 \font\eightpoint=cmr8 \def\rmi{{\rm i}} \def\square{\mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3} \def\sup{\mathop{\rm sup}} \def\Dom{D} \def\ve{\varepsilon} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \hbox to \hsize{\hfil\eightit 3/August/1996} \vglue1truein %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\bf OPTIMAL HEAT KERNEL ESTIMATES FOR} \centerline{\bf SCHR\"ODINGER OPERATORS WITH MAGNETIC FIELDS} \centerline{\bf IN TWO DIMENSIONS} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \bigskip \bigskip {\baselineskip=2.5ex \vfootnote{}{\eightpoint \noindent To appear in: Commun. Math. Phys.\medskip}} {\baselineskip=2.5ex \vfootnote{}{\eightpoint \noindent\copyright 1996 by the authors. Reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes.}} {\baselineskip = 12pt \halign{\qquad\qquad#\hfil\qquad\qquad\qquad\qquad\quad\hfil&#\hfil\cr {\bf Michael Loss}\footnote {$^*$}{Work supported by N.S.F. grant DMS-95-00840 and the Erwin Schr\"odinger Institute} & {\bf Bernd Thaller}\cr &\cr School of Mathematics & Institut f\"ur Mathematik\cr Georgia Institute of Technology & Universit\"at Graz\cr Atlanta, GA 30332, USA & A-8010 Graz, Austria\cr}} \bigskip \vskip .7 true in \centerline{\bf Abstract} \smallskip {\rightskip=4pc \leftskip=4pc \tenrm\baselineskip=11pt\parindent=1pc Sharp smoothing estimates are proven for magnetic Schr\"odinger semigroups in two dimensions under the assumption that the magnetic field is bounded below by some positive constant $B_0$. As a consequence the $L^{\infty}$ norm of the associated integral kernel is bounded by the $L^{\infty}$ norm of the Mehler kernel of the Schr\"odinger semigroup with the constant magnetic field $B_0$.\par} \noindent {\vglue 0.5cm} \vfill\eject %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent{\bf 1. INTRODUCTION} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent A recurring problem in the theory of semigroups is to prove smoothing estimates, i.e., estimates of the semigroup acting as an operator from $L^p \to L^q$, $q \geq p$. (see [S82] for a general review on semigroups) In rare cases it is possible to render these estimates in their sharp form. Among the examples are Nelson's hypercontractive estimate [N66] and the sharp smoothing properties of the heat kernel (which follow from the sharp form of Young's inequality [B75], [BL76]). A more recent example is furnished by the Mehler kernel associated with the Schr\"odinger operator of a charged particle in a constant magnetic field which essentially is a result of Lieb [L90] although not explicitly stated there (see Theorem~1.1). An especially flexible approach to such problems, due to Gross [G76] (see also Federbush [F69]) and relevant for our paper, is the use of logarithmic Sobolev inequalities. In fact there is complete equivalence between Gross' logarithmic Sobolev inequality and Nelson's hypercontractive estimate. In a similar fashion, Weissler [W78] proved the equivalence of the sharp smoothing estimates for the heat semigroup and logarithmic Sobolev inequalities. On a more abstract level it was realized in Simon and Davies [DS84] (see also [D89]) that the technique of logarithmic Sobolev inequalities can be used to prove ultracontractivity for Markov semigroups. It is the aim of this note to implement Gross' method for magnetic Schr\"odinger operators and to prove sharp smoothing estimates for the associated semigroup. The idea is similar to the one in [CL96] where it was realized that Gross' logarithmic Sobolev inequality when viewed as a family of sharp inequalities on $\Bbb R^n$ can be used to obtain sharp smoothing estimates of solutions of diffusion equations with a volume preserving drift. An example is the two dimensional Navier--Stokes equations in the vorticity formulation. Consider a single charged quantum particle in a magnetic field, i.e., consider the Hamiltonian % $$H(B)={1 \over 2}(\nabla+\rmi A)^2 \ , \eqno(1.1)$$ % where $A$ is a vector potential for the magnetic field $B$, i.e., ${\rm curl}\, A= B$. Note that the semigroup generated by (1.1) is not Markovian, in fact not even positivity preserving. Usually, this difficulty is overcome by the important diamagnetic inequality ([K72], [S77,79], [HSU77], see also [AHS77] and [CFKS87]) % $$\left|\bigl(e^{tH}u\big)(x)\right| \le \bigl ( e^{t\Delta/2}|u| \bigr)(x)\ , \eqno(1.2)$$ % which relates estimates on magnetic Schr\"odinger operator semigroup to estimates on the heat semigroup. The obvious disadvantage of using (1.2) is that all effects due to the magnetic field are completely eliminated. How, then, does the magnetic field affect the behavior of the semigroup? The first paper that addressed this question is presumably [M86]. It is proved in [M86] that for a magnetic Schr\"odinger operator in ${\Bbb R}^3$ $$\lim_{t \to \infty}{1 \over t}\,{\rm ln}\, \Vert e^{tH(B)} \Vert_{L^1 \to L^{\infty}} \leq - \Phi (C) B_0 \ , \eqno(1.3)$$ for all curl-free magnetic fields $B$ that satisfy $0 < B_0 \leq |B| \leq C B_0$, Here $|B|=\sqrt {B_1^2+B_2^2+B_3^2}$ with $B_1,B_2,B_3$ being the components and the function $\Phi$ is explicitly given in [M86]. It is noteworthy that the estimates were obtained by probabilistic techniques. This work was subsequently improved in [E94]. The estimate (1.3) holds in any dimension and without the condition that $B$ be curl-free. In particular the following stronger estimate is true $$\lim_{t \to \infty}{1 \over t}\,{\rm ln}\, \Vert e^{tH(LB)} \Vert_{L^1 \to L^{\infty}} \leq - C_L\min_{x \in\Bbb R^n}|B(x)|L \ .$$ Here the constant $C_L=1+O(L^{\alpha}\,{\rm ln}\, L)$ as $L \to \infty$ for some explicitly given $\alpha<0$. Thus, this result is in a certain sense optimal since by considering a constant magnetic field in two dimensions one cannot improve the {\it exponent}\/ $\min_{x \in\Bbb R^n}|B(x)|$. Again the results in [E94] were obtained by probabilistic techniques. Similar results can be found in [U94] where techniques from differential geometry were used. \medskip In this paper we consider the simpler problem for a magnetic field of constant direction, or what amounts to the same, a magnetic field in two dimensions. However, we try to retain as much information as possible about the magnetic field in the estimate on the semigroup. A magnetic field in two dimensions is a scalar function $B$ which can be expressed in terms of a (non-unique) vector potential $A$, % $$B(x) = {\partial A_2(x)\over \partial x_1} - {\partial A_1(x)\over \partial x_2}\ , \qquad A(x) = \bigl(A_1(x),A_2(x)\bigr)\ .$$ % For example, if $B(x)=B_0$ is a constant magnetic field then % $$A(x) = {B_0\over 2}\,\bigl(-x_2,x_1\bigr)\ . \eqno(1.4)$$ % We want to estimate (for suitable $p$ and $q$ with $1\le p\le q \le \infty$) % $$C(t;p,q)\,:=\,\sup_{u\in L^p}\,{\|u(t)\|_q \over \|u\|_p} \qquad\hbox{where}\quad u(t) = e^{tH}\,u\ ,$$ % which is just the norm of the Schr\"odinger semigroup $\exp(tH)$ as a mapping from $L^p({\Bbb R}^2)$ to $L^q({\Bbb R}^2)$. This norm is finite, because (1.2) implies that % $$\| e^{tH} \|_{L^p\to L^q} \le \|e^{t\Delta/2}\|_{L^p\to L^q} < \infty \ .$$ % In case of a constant magnetic field $B(x)=B_0>0$ these calculations can be done using the explicitly known integral kernel of the time evolution operator. If the vector potential $A$ is chosen as in Eq.~(1.4) above, the time evolution is given by % $$u(t,x) = \int G(t,x,y)\,u(y)\,d^2\!y \ ,$$ % where $G(t,x,y)$ is the Mehler kernel [S79, p.~168] % $$G(t,x,y) = {B_0\over 4}\,{1\over \pi \sinh\Bigl({B_0 t\over 2}\Bigr)}\, \exp\left\{ -{B_0\over 4}\,\coth\Bigl({B_0 t\over 2}\Bigr)\,(x-y)^2 +\rmi\,{B_0\over 2}\,(x_1y_2 -x_2y_1)\right\} \ ,$$ % which is a complex, degenerate, centered Gaussian kernel (in the terminology of [L90]). In particular, we need the time evolution of the initial function % $$u_0(x) = N_0\,\exp\left\{-{B_0\over 4a_0}\,x^2\right\},\qquad a_0>0 \ . \eqno(1.5)$$ % The solution at time $t$ is again a Gaussian which can be written as % $$u_0(t,x) = N(t)\,\exp\left\{-{B_0\over 4a(t)}\,x^2\right\} \ ,$$ % where % \def\SH{\sinh\left({B_0 t\over 2}\right)} \def\CH{\cosh\left({B_0 t\over 2}\right)} $$N(t) = {N_0\,a_0\over a_0\,\CH + \SH} \ , \qquad a(t) = {a_0\,\CH + \SH\over a_0\,\SH + \CH} \ .$$ % The function $a(t)$ is the unique solution of the initial value problem % $$\dot a(t) = {B_0\over 2} \bigl(1-a(t)^2\bigr) \ ,\qquad a(0) = a_0 \ .$$ % Writing $a_0 = \tanh(\theta)$ (for $a_0<1$), or $a_0=\coth(\theta)$ (for $a_0>1$), we obtain % $$a(t) = \cases{\tanh\left(\theta + {B_0 t\over 2}\right)& for a_0<1,\cr 1 & for a_0=1,\cr \coth\left(\theta + {B_0 t\over 2}\right)& for a_0>1.} \eqno(1.6)$$ % For $a_0 = 1$, the function $u_0(x) = N_0\exp(-B_0x^2/4)$ is the eigenfunction of $H$ corresponding to the eigenvalue $-B_0/2$. The following theorem sets the stage for our investigation. It is essentially a corollary of a theorem in [L90]. We shall give the details in Sect.~2. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent {\bf Theorem 1.1: } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Let $B(x) = B_0>0$, all $x\in{\Bbb R}^2$. For $1q$. {\bf Remark 2:} Instead of using the results of [L90] one might approach Theorem 1.1 a la Gross [G76],i.e., trying to reduce the problem to an integral over infinitesimal time steps, and estimating the semigroup over these time steps by employing logarithmic Sobolev inequalities. It is clear that one would get some smoothing estimates but can one obtain them in the sharp form? That there are some obstructions to reach this goal by this method can be seen as follows. Eqs.~(1.5) and (1.7) determine the Gaussian function $u_0$ that yields the norm of the magnetic heat kernel as an operator from $L^p$ to $L^q$. With this Gaussian we may write % $$C_0(t;p,q) = {\|u_0(t)\|_q\over \|u_0\|_p} = {\|u_0(s)\|_r\over \|u_0\|_p}\, {\|u_0(t)\|_q\over \|u_0(s)\|_r}\ ,$$ % for some $r$ and $s$ with $p\le r\le q$, $0\le s\le t$. Hence it is obvious that % $$C_0(t;p,q) \le C_0(s;p,r)\,C_0(t-s,r,q)\ .$$ % Clearly, the proposed method can only work, if there exists a number $r$ with $p 2$, $00$ is continuous. Then the estimate % $$C(t;p,q) = \|e^{tH(B)}\|_{L^p\to L^q} \le C_0(t;p,q)$$ % (with $C_0(t;p,q)$ given by Eq.~(1.8)) holds for \smallskip\noindent {\bf a}) all $t>0$, if $p\le 2$, $q\ge 2$, \smallskip\noindent {\bf b}) $t>0$ and $20$ and $1\le p0$ is continuous. Then the magnetic heat kernel satisfies the bound % $$|e^{tH}(x,y)| \leq {B_0 \over 4 \pi \sinh({B_0 t \over 2})} e^{-{(x-y)^2 \over 2t}} \ . \eqno(1.14)$$ % } \medskip {\bf Remark:} The Gaussian decay on the right side of (1.14) is the one of the heat kernel, which is considerably weaker than the decay of the Mehler kernel. Is it true that $|e^{tH}(x,y)|$ is bounded by the Mehler kernel $|G(t,x,y)|$? The truth of this estimate would reveal a robust dependence of the magnetic heat kernel on the magnetic field. This is an open problem. \medskip {\bf Acknowledgment:} We would like to thank Eric Carlen, Laszlo Erd\"os, and Vitali Vugalter for many helpful discussions. \bigskip\bigskip\bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent{\bf 2. PROOF OF THEOREM 1.1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent Let $G_t$ be the integral operator with the Mehler kernel $G(t,x,y)$. We want to determine % $$\sup_u {\| G_t u\|_q \over \|u\|_p} = C_0(t;p,q),$$ % which is either finite or infinite. By a result of E.~Lieb (see [L90], Theorem~4.1) it is sufficient to take the supremum over centered Gaussian functions of the form % $\exp( - x\cdot J_\ve x )$, % where $J_\ve$ is a complex-valued matrix, symmetric with respect to the scalar product in ${\Bbb R}^2$, and with a strictly positive real part. Even for this case the calculation is still somewhat tedious. Instead it is convenient to rework the proof of Theorem~4.1 in [L90]. Let $t>0$ be arbitrary. We approximate $G_t$ by the operator $G_t^\ve$ with kernel % $$G_t^\ve (x,y) = {\rm e}^{-\ve x^2}\,G(t,x,y),$$ % which for $\ve>0$ is a non-degenerate, centered Gaussian kernel. According to [L90], Theorem~3.4, there is a unique (up to a multiplicative constant) centered Gaussian function $u_\ve$ which is the maximum of $\| G_t^\ve u\|_q/\|u\|_p$ over all $u\in L^p({\Bbb R}^2)$. The function $u_\ve$ is of the form % $u_\ve(x) = \exp( - x\cdot J_\ve x )$, % where $J_\ve$ is a (possibly complex-valued) matrix which is symmetric with respect to the scalar product in ${\Bbb R}^2$ and has a strictly positive real part. But since the integral operator $G_t^\ve$ commutes with rotations, the unique maximum $u_\epsilon$ must also be rotationally invariant. Hence $J_\ve= \alpha_\ve {\bf 1} + {\rm i}\beta_\ve {\bf 1}$, where $\alpha_\ve>0$, and $\beta_\ve$ is real. Since the integrals of Gaussian functions can be evaluated explicitly, we can evaluate the quotient $\|G_t^\ve u\|_q/\|u\|_p$ for $u(x)=\exp[-(\alpha + \rmi \beta)\,x^2]$ and maximize this expression over all $\alpha$ and $\beta$. The maximum is obtained for $(\alpha,\beta)=(\alpha_\ve,\beta_\ve)$, with $\alpha_\ve>0$ and $\beta_\ve = 0$. Since $\exp(-\ve x^2)\le 1$ we find for any Gaussian function $u$ % $$\|G_t^\ve u\|_q \le \|G_tu\|_q$$ % and hence % $$C^\ve={\|G_t^\ve u_\ve\|_q\over \|u_\ve\|_p} \le C_0(t;p,q),$$ % where $u_\ve = \exp(-\alpha_\ve x^2)$ is the unique Gaussian maximizer for $G_t^\ve$. By an explicit calculation one sees that % $$\lim_{\ve \to 0} C^\ve \equiv C^0 = {\| G_t u_0 \|_q \over \|u_0\|_p},$$ % with $u_0(x) = \exp(-\alpha x^2)$, $\alpha = \lim_{\ve\to 0} \alpha_\ve$. Of course, we have $C^0\le C_0(t;p,q)$. Finally, for any Gaussian function $u$, the limit % $$\lim_{\ve \to 0 } {\|G_t^\ve u\|_q\over \|u\|_p} = {\| G_tu\|_q\over \|u\|_p}$$ % exists by Lebesgue's dominated convergence, and from % $${\|G_t^\ve u\|_q\over \|u\|_p} \le C^\ve \quad \hbox{for all \ve > 0}$$ % we find immediately that % $${\| G_t u\|_q\over \|u\|_p} \le \lim_{\ve \to 0} C^\ve = C^0 \quad\hbox{for all Gaussian functions u}.$$ % Hence, also $\sup \|G_t u\|_q/\|u\|_p = C_0(t;p,q) \le C^0$. This proves $C^0 = C_0(t;p,q)$. A little calculation easily gives the explicit value of this constant. \vfill\eject %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent{\bf 3. A DIFFERENTIAL INEQUALITY} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent In the following we assume that the magnetic field $B$ is given by a differentiable vector field $A$ as in Eq.~(1.1) and satisfies $B(x) \ge B_0 > 0$, for all $x \in {\Bbb R}^2$. In particular we choose the vector potential to be in $L^4_{\rm loc}(\Bbb R^2)$ and then by the Leinfelder--Simader Theorem [LS81] the formal expression (1.1) defines a selfadjoint operator on some domain $\Dom(H)$ with $C^{\infty}_0(\Bbb R^2)$ as a core. If we set $u(s)=e^{Hs}u_0$ then $u(s)$ is a solution of % $${d\over ds}\,u = H\,u={1 \over 2}(\nabla + \rmi A)^2\,u \ . \eqno(3.1)$$ % If $u_0 \in \Dom(H) \cap L^1(\Bbb R^2) \cap L^{\infty}(\Bbb R^2)$ then we have that $u(s)\in \Dom(H) \cap L^1(\Bbb R^2) \cap L^{\infty}(\Bbb R^2)$. This follows from the diamagnetic inequality (1.2) for each $s\in (0,t]$ and the explicit form of the heat kernel. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent {\bf Theorem 3.1: } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\it Let $r:(0,t]\to {\Bbb R}$ be a twice differentiable function with $r(s) \geq 2$ and $\dot r(s) \ge 0$ for all $s\in (0,t]$. Then % $${d\over ds}\,\ln \|u(s)\|_{r(s)} \le - L(r(s),\dot r(s)) \ , \eqno(3.2)$$ % for all $s\in (0,t]$, where % $$L(r, \dot r) = {\dot r\over r^2}\,\left\{ 2+\ln{4\pi a(r,\dot r)\over B_0 r} \right\} + {a(r,\dot r) B_0\over r} \ . \eqno(3.3)$$ % and % $$a(r,\dot r) = {1\over B_0 r}\bigl( \sqrt{\dot r^2 + B_0^2 r^2 (r-1)} - \dot r \bigr).$$ % Equality holds in Eq~(3.2) for $B(x) = B_0$ and % $$u(s,x) = \exp\left( - {B_0\over 4 a\bigl(r(s),\dot r(s)\bigr)} \,x^2 \right) \ .$$ % } \medskip\noindent {\it Proof\/}: Pick $u_0 \in \Dom(H) \cap L^1(\Bbb R^2) \cap L^{\infty}(\Bbb R^2)$. Since ${d\over d\tau}\,\ln k\|u(\tau)\|_{r(\tau)}$ does not depend on $k>0$, we assume during the following calculation, without loss of generality, that at the time $\tau=s>0$ the solution is normalized such that $\|u(s)\|_{r(s)} = 1$. For the derivative at $\tau=s$ we obtain therefore % \eqalign{{d\over ds}\,\ln \|u(s)\|_{r(s)} = & \,{\dot r(s)\over r(s)^2}\int | u(s,x) |^{r(s)} \ln | u(s,x) |^{r(s)}\,d^2\!x \cr & + {1 \over 2}\int |u(s,x)|^{(r(s)-2)}\, {d\over ds}|u(s,x)|^2 \,d^2\!x \ .\cr} \eqno(3.4) % The integrals have to be taken over ${\Bbb R}^2$. The formal computation can be easily justified by an approximation argument. For simplicity, the arguments $s$ and $x$ in the integrand on the right side will be omitted from now on. Using (3.1) we obtain, after a partial integration % \eqalign{{1\over 2}\int |u|^{(r-2)}\,{d\over ds} |u|^2 = & -{1 \over 2}(r-2)\int|u|^{(r-2)}\,(\nabla |u|)^2\cr & -{1\over 2}\int |u|^{(r-2)}\,|(\nabla + \rmi A)\,u|^2 \ .} \eqno(3.5) % The integration by parts can be justified as follows. Since $u \in \Dom(H)$, by the Leinfelder--Simader Theorem [LS81] we can pick a sequence $u_n \in C^{\infty}_0 ({\Bbb R}^2)$ such that $u_n \to u, Hu_n \to Hu$ in $L^2$. Inspecting the proof of the Leinfelder--Simader Theorem one sees that the sequence $u_n$ can be chosen to converge to $u$ in $L^1$ and to have a uniform bound on the $L^{\infty}$ norm. Thus $u_n$ converges to $u$ in $L^p$ for all $1\leq p <\infty$. In particular, all the following computations can be justified in the same fashion and we can assume without restriction that $u \in C^{\infty}_0({\Bbb R}^2)$. If we set $u=f+\rmi g$ then $|u|= \sqrt {f^2+g^2}$. We find % $$|(\nabla + \rmi A)\,u|^2= (\nabla |u|)^2 + |A+\nabla S|^2\,|u|^2 = X^2+Y^2 \eqno(3.6)$$ % where we have introduced two real vector fields $X$ and $Y$ over ${\Bbb R}^2$, % $$X = \nabla^\bot |u| = \Bigl({\partial\over \partial x_2},-{\partial\over \partial x_1}\Bigr)|u|,$$ % $$Y = (A+\nabla S)\,|u|.$$ % Here the symbol $\nabla S$ denotes the expression % $${f \nabla g - g \nabla f \over f^2+g^2} ,$$ % which is defined wherever $f^2+g^2 > 0$. For any $c>0$ we may estimate % $$c^2X^2 + Y^2 \ge 2c \, X \cdot Y \eqno(3.7)$$ % with equality if and only if $cX=Y$. If $B=B_0$ is a constant magnetic field we can choose a Gaussian function % $$u=N\,\exp\left\{-{B_0\over 4c}\,x^2\right\} \eqno(3.8)$$ % such that equality holds everywhere in (3.7), because (with $A$ as in Eq.~(1.2)) % $$cX = c\,\nabla^\bot u = {B_0\over 2}\,(-x_2,x_1)\,u = Au = Y \ .$$ % Let us now insert (3.6) and (3.7) into (3.5). Since $r-1>0$ we can add and subtract a positive constant $c^2$ with $0 0$. Using $B(x)\ge B_0$ and $\int |u|^{r}=1$ these estimates lead to the following family of inequalities for each $c\in (0,\sqrt{r-1})$ % \eqalign{{d\over ds}\,\ln \|u\|_{r} &\le {\dot r\over r^2} \int |u|^{r} \ln |u|^{r} \cr &- {1\over 2} (r-1-c^2)\int |u|^{(r-2)} (\nabla|u|)^2 -{c B_0\over r} \ .\cr} \eqno(3.9) % Equality holds in case of a constant magnetic field $B=B_0$ and $u$ a Gaussian function given by Eq.~(3.8). We also note that in the special case $\dot r(s) = 0$ the above calculation can be simplified to give % $${d\over ds}\,\ln \|u\|_{r} \le -\sqrt{r-1}\,{B_0\over r} \ . \eqno(3.10)$$ Now we apply the following family of logarithmic Sobolev inequalities % $$\int g^2\ln g^2 - {\lambda\over \pi}\int(\nabla g)^2 \le -2-\ln\lambda \ ,\qquad \hbox{all \lambda > 0}, \eqno(3.11)$$ % where we assumed that $\int g^2 = 1$. Equality holds if $g$ is a normalized Gaussian, % $$g(x) = {1\over\sqrt{\lambda}}\,\exp\left\{ -{\pi\over 2\lambda}\,x^2 \right\} \ .$$ % This family is essentially Gross' logarithmic Sobolev inequality rewritten as an inequality on ${\Bbb R}^2$ (see, e.g., [CL96]). We insert (3.11) into (3.9) with % $$\lambda = {2\pi\over \dot r} (r-1-c^2),\qquad g = |u|^{r/2} \ . \eqno(3.12)$$ % Here we have to assume that $\dot r>0$. With this choice of $\lambda$ we obtain % $${d\over ds}\,\ln \|u\|_{r} \le -{\dot r\over r^2} \left\{ 2+\ln{2\pi\over\dot r}(r-1-c^2) \right\} -{cB_0\over r} \ . \eqno(3.13)$$ % This result holds for all \$0