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2. Heat-smoothing conjecture

In this part, we present some problems related to the heat-smoothing conjecture. See [MR3210176], [MR3632552] and [arXiv:2011.01359] for more informaiton.

    1. Heat-smoothing conjecture


      Add remark

      Problem 2.1.

      Fix $d\ge 1$ and $1< p < \infty$. Show that there exists $c_p>0$ such that for all $n\ge 1$ and all $f:\{-1,1\}^n\to \mathbb{R}$ of the form $f=\sum_{|S|\ge d}\widehat{f}(S)\chi_S$, we have \begin{equation*} \|P_t f\|_p\le e^{-tdc_p}\|f\|_p, \qquad t\ge 0, \end{equation*} where $P_t=e^{-t\Delta}$ is the heat semigroup on $\{-1,1\}^n$.

        • Heat-smoothing: a weaker form


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          Problem 2.2.

          [Paata Ivanisvili] Fix $d\ge 1$ and $1< p < \infty$. Show that there exists $c_p>0$ such that for all $n\ge 1$ and all $f:\{-1,1\}^n\to \mathbb{R}$ of the form $f=\sum_{|S|\ge d}\widehat{f}(S)\chi_S$, we have \begin{equation*} \|\Delta f\|_p\ge dc_p\|f\|_p, \end{equation*} where $\Delta$ is the generator of the heat semigroup $P_t=e^{-t\Delta}$.

            • Gaussian heat-smoothing

                  One may also consider the Gaussian analogs, which are weaker using a limit argument. Here $\gamma_n$ is the standard Gaussian probability measure on $\mathbb{R}^n$ and we denote $\|\cdot\|_p:=\|\cdot\|_{L_p(\mathbb{R}^n,\gamma_n)}$.

              Add remark

              Problem 2.3.

              Fix $d\ge 1$ and $1< p < \infty$. Show that there exists $c_p>0$ such that for all $n\ge 1$ and all polynomials $f\in \text{Lin}\{x^k,k\ge d\}$, we have \begin{equation*} \|P_t f\|_p\le e^{-tdc_p}\|f\|_p, \qquad t\ge 0, \end{equation*} where $P_t=e^{-tN}$ is the Ornstein–Uhlenbeck semigroup on the Gaussian space.
                1. Remark. [Paata Ivanisvili] in Problem 2.3. it should be that f belongs to Lin{H_k, |k| \geq d}, where H_k are Hermite polynomials indexed by multiindeces k.

                    • Gaussian heat-smoothing: a weaker form

                          Similarly, we have the following weaker form of Gaussian heat-smoothing, with the same convention as in the previous problem.

                      Add remark

                      Problem 2.4.

                      [Paata Ivanisvili] Fix $d\ge 1$ and $1< p < \infty$. Show that there exists $c_p>0$ such that for all $n\ge 1$ and all polynomials $f\in \text{Lin}\{x^k,k\ge d\}$, we have \begin{equation*} \|Nf\|_p\ge dc_p\|f\|_p, \end{equation*} where $N=\Delta-x\nabla$ is the generator of the Ornstein–Uhlenbeck semigroup.
                        1. Remark. [Paata Ivanisvili] It should be that f belongs to Lin{H_k, |k|\geq d}, where H_k are hermite polynoimals.

                            • Reverse heat-smoothing

                                  One may also consider the above inequalities for polynomials of low degree.

                              Add remark

                              Problem 2.5.

                              [Paata Ivanisvili] Fix $d\ge 1$ and $1< p < \infty$. Show that there exists $c_p>0$ such that for all $n\ge 1$,
                              1. and for all polynomials $f$ on $\{-1,1\}^n$ of degree at most $d$, we have \begin{equation*} \|\Delta f\|_p\le dc_p\|f\|_p \end{equation*} and thus \begin{equation*} \|P_t f\|_p\le e^{tdc_p}\|f\|_p, \qquad t\ge 0, \end{equation*} where $P_t=e^{-t\Delta}$ is the heat semigroup on $\{-1,1\}^n$;
                              2. and for all polynomials $f$ on $\mathbb{R}^n$ of degree at most $d$, we have \begin{equation*} \|N f\|_p\le dc_p\|f\|_p \end{equation*} and thus \begin{equation*} \|P_t f\|_p\le e^{tdc_p}\|f\|_p, \qquad t\ge 0, \end{equation*} where $P_t=e^{-tN}$ is the Ornstein–Uhlenbeck semigroup on $(\mathbb{R}^n,\gamma_n)$.
                                1. Remark. [Paata Ivanisvili] One remark is that these two inequalities(one with Laplacian and one with the semigroup) imply one another.

                                      Cite this as: AimPL: Analysis on the hypercube with applications to quantum computing, available at http://aimpl.org/hypercubequantum.