An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family

<jats:p>Let <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mi>X</mi>
                     </math>
                  </jats:inline-formula> be a topological space equipped with a complete positive <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>σ</mi>
                     </math>
                  </jats:inline-formula>-finite measure and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>T</mi>
                     </math>
                  </jats:inline-formula> a subset of the reals with <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mn>0</mn>
                     </math>
                  </jats:inline-formula> as an accumulation point. Let <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <msub>
                           <mrow>
                              <mi>a</mi>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>x</mi>
                              <mo>,</mo>
                              <mi>y</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> be a nonnegative measurable function on <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>X</mi>
                        <mo>×</mo>
                        <mi>X</mi>
                     </math>
                  </jats:inline-formula> which integrates to <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mn>1</mn>
                     </math>
                  </jats:inline-formula> in each variable. For a function <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
                        <mi>f</mi>
                        <mo>∈</mo>
                        <msub>
                           <mrow>
                              <mi>L</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>X</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
                        <mi>t</mi>
                        <mo>∈</mo>
                        <mi>T</mi>
                     </math>
                  </jats:inline-formula>, define <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
                        <msub>
                           <mrow>
                              <mi>A</mi>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msub>
                        <mi>f</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>x</mi>
                           </mrow>
                        </mfenced>
                        <mo>≡</mo>
                        <mo>∫</mo>
                        <mo> </mo>
                        <msub>
                           <mrow>
                              <mi>a</mi>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>x</mi>
                              <mo>,</mo>
                              <mi>y</mi>
                           </mrow>
                        </mfenced>
                        <mi>f</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>y</mi>
                           </mrow>
                        </mfenced>
                        <mtext> </mtext>
                        <mi>d</mi>
                        <mi>y</mi>
                     </math>
                  </jats:inline-formula>. We assume that <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
                        <msub>
                           <mrow>
                              <mi>A</mi>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msub>
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula> converges to <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula> in <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
                        <msub>
                           <mrow>
                              <mi>L</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msub>
                     </math>
                  </jats:inline-formula>, as <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14">
                        <mi>t</mi>
                        <mo>⟶</mo>
                        <mn>0</mn>
                     </math>
                  </jats:inline-formula> in <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15">
                        <mi>T</mi>
                     </math>
                  </jats:inline-formula>. For example, <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16">
                        <msub>
                           <mrow>
                              <mi>A</mi>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msub>
                     </math>
                  </jats:inline-formula> is a diffusion semigroup (with <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17">
                        <mi>T</mi>
                        <mo>=</mo>
                        <mfenced open="[" close=")">
                           <mrow>
                              <mn>0</mn>
                              <mrow>
                                 <mo>,</mo>
                              </mrow>
                              <mrow>
                                 <mo>∞</mo>
                              </mrow>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>). For <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M18">
                        <mi>W</mi>
                     </math>
                  </jats:inline-formula> a finite measure space and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M19">
                        <mi>w</mi>
                        <mo>∈</mo>
                        <mi>W</mi>
                     </math>
                  </jats:inline-formula>, select real-valued <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M20">
                        <msub>
                           <mrow>
                              <mi>h</mi>
                           </mrow>
                           <mrow>
                              <mi>w</mi>
                           </mrow>
                        </msub>
                        <mo>∈</mo>
                        <msub>
                           <mrow>
                              <mi>L</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>X</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>, defined everywhere, with <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M21">
                        <msub>
                           <mrow>
                              <mfenced open="‖" close="‖">
                                 <mrow>
                                    <msub>
                                       <mrow>
                                          <mi>h</mi>
                                       </mrow>
                                       <mrow>
                                          <mi>w</mi>
                                       </mrow>
                                    </msub>
                                 </mrow>
                              </mfenced>
                           </mrow>
                           <mrow>
                              <msub>
                                 <mrow>
                                    <mi>L</mi>
                                 </mrow>
                                 <mrow>
                                    <mn>2</mn>
                                 </mrow>
                              </msub>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mi>X</mi>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </msub>
                        <mo>≤</mo>
                        <mn>1</mn>
                     </math>
                  </jats:inline-formula>. Define the distance <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M22">
                        <mi>D</mi>
                     </math>
                  </jats:inline-formula> by <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M23">
                        <mi>D</mi>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>x</mi>
                              <mo>,</mo>
                              <mi>y</mi>
                           </mrow>
                        </mfenced>
                        <mo>≡</mo>
                        <msub>
                           <mrow>
                              <mfenced open="‖" close="‖">
                                 <mrow>
                                    <msub>
                                       <mrow>
                                          <mi>h</mi>
                                       </mrow>
                                       <mrow>
                                          <mi>w</mi>
                                       </mrow>
                                    </msub>
                                    <mfenced open="(" close=")">
                                       <mrow>
                                          <mi>x</mi>
                                       </mrow>
                                    </mfenced>
                                    <mo>−</mo>
                                    <msub>
                                       <mrow>
                                          <mi>h</mi>
                                       </mrow>
                                       <mrow>
                                          <mi>w</mi>
                                       </mrow>
                                    </msub>
                                    <mfenced open="(" close=")">
                                       <mrow>
                                          <mi>y</mi>
                                       </mrow>
                                    </mfenced>
                                 </mrow>
                              </mfenced>
                           </mrow>
                           <mrow>
                              <msub>
                                 <mrow>
                                    <mi>L</mi>
                                 </mrow>
                                 <mrow>
                                    <mn>2</mn>
                                 </mrow>
                              </msub>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mi>W</mi>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </msub>
                     </math>
                  </jats:inline-formula>. Our main result is an equivalence between the smoothness of an <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M24">
                        <msub>
                           <mrow>
                              <mi>L</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>X</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> function <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M25">
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula> (as measured by an <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M26">
                        <msub>
                           <mrow>
                              <mi>L</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msub>
                     </math>
                  </jats:inline-formula>-Lipschitz condition involving <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M27">
                        <msub>
                           <mrow>
                              <mi>a</mi>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mo>·</mo>
                              <mo>,</mo>
                              <mo>·</mo>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> and the distance <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M28">
                        <mi>D</mi>
                     </math>
                  </jats:inline-formula>) and the rate of convergence of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M29">
                        <msub>
                           <mrow>
                              <mi>A</mi>
                           </mrow>
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </msub>
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula> to <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M30">
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula>.</jats:p>


Introduction
One of the questions that arise in harmonic analysis is the connection between the smoothness of a given function and the rate of approximation by members of a specified family of functions. An important example is the relationship between the smoothness of a function and the speed of convergence of its diffused version to itself, in the limit as time goes to zero. As mentioned in the Introduction of [1], for the Euclidean setting and the heat kernel, see for example [2,3].
In a more general setting, for a diffusion semigroup fT t f g t≥0 on a topological space X with a positive σ-finite measure given, for t > 0, by an integral kernel operator: T t f ðxÞ ≡ Ð X ρ t ðx, yÞf ðyÞ dy, Coifman and Leeb in [1,4] introduce a family of multiscale diffusion distances and establish quantitative results about the equivalence of a bounded function f being Lipschitz and the rate of convergence of T t f to f , as t → 0 + . The respective authors of [5][6][7] consider different aspects of the connection between the smoothness of a function and the rate of convergence of its diffused versions to itself.
As mentioned in, for instance, the Introductions of [5][6][7], the interest in diffusion semigroups is natural since they play an important role in analysis, both theoretical and applied. Diffusion semigroups include the heat semigroup and, more generally, as discussed in, e.g., [8], arise from considering large classes of elliptic second-order (partial) differential operators on domains in Euclidean space or on manifolds.
In the present work, we consider a more general family than a diffusion semigroup. For T a subset of the reals having 0 as an accumulation point, for t ∈ T, let a t ðx, yÞ be a nonnegative measurable function on X × X which integrates to 1 in each variable. For a function f ∈ L 2 ðXÞ and t ∈ T, define A t f ðxÞ = Ð X a t ðx, yÞf ðyÞ dy. We assume that for every f ∈ L 2 ðXÞ, No assumption is made that the family A t is symmetric or is a semigroup nor is anything assumed about T other than that T has 0 as an accumulation point.
For a finite measure space W, selecting h w ∈ L 2 ðXÞ for every w ∈ W, we define a distance between points x, y ∈ X by Dðx, yÞ ≜ kh w ðxÞ − h w ðyÞk L 2 ðWÞ . We next introduce an L 2 version of being Lipschitz (relative to fA t g) using this distance D. Our main result is that a function f ∈ L 2 ðXÞ is L 2 -Lipschitz if and only if we have an estimate of the rate of Our paper is organized as follows. Following a notation and assumptions section (Section 2), we state the main definitions, provide some examples, and establish our results in Section 3. The paper ends with the Conclusions and Acknowledgments sections.

Notation and Assumptions
Let X be a topological space equipped with a complete positive σ-finite measure. The measure on X will be denoted by dx and dy. W is a finite measure space, with measure denoted by dw. We assume all spaces involved are such that Fubini's theorem holds on any product of these spaces; e.g., the spaces are σ-finite. All functions are assumed to be real-valued and measurable on the respective spaces; in particular, functions of several variables are assumed to be measurable on the appropriate product spaces.
T will denote a subset of the reals, with 0 as an accumulation point. From now on, t ⟶ 0 will mean t ⟶ 0, t ∈ T. For every t ∈ T, let a t ðx, yÞ be a nonnegative measurable function on X × X with the property that Ð X a t ðx, yÞ dy = Ð X a t ðx, yÞ dx = 1. For t ∈ T and a function f ∈ L 2 ðXÞ, define No assumption is made that the family A t is symmetric or is a semigroup nor is anything assumed about T other than that T has 0 as an accumulation point.
Note that A t is indeed bounded on L 2 with norms not exceeding one, since In We will define a family Δ of symmetric distances on X × X satisfying the triangle inequality with the following properties for every D ∈ Δ:

Main Definitions and Results
We start by describing the family Δ of symmetric distances on X × X.
Note that some h w may be chosen to be identically 0. Then, the distance D ∈ Δ is given by Clearly, D is symmetric and satisfies the triangle inequality (the latter fact follows from the triangle inequality for L 2 ðWÞ).
Before looking at some examples of such distances, we define our L 2 -Lipschitz condition.
for every t ∈ T. Now let us consider some examples of distances D ∈ Δ. For the first one, let X be a bounded subset of ℝ n having (some) finite measure dx. Let W = f1, 2, ⋯, ng, with dw indicating unit masses assigned at each point of W. for the basic case when X = ℝ n with dx Lebesgue measure and fA t g is the heat flow semigroup. (While the derivation right after the statement of Proposition 2.6 in [5] by Coifman and Goldberg has a calculation of this distance, we present a more detailed computation here. ) We easily see that 2e −jx−yj 2 /ð16sÞ , where |x − y | is the Euclidean distance between the points x and y. Thus, D 2 2 ðx, yÞ = 2 If |x − y | ≥1, e −jx−yj 2 /ð16sÞ is bounded away from 1 for 0 < s < 1, so D 2 2 ðx, yÞ~Ð 1 0 s α−1 ds = c. If |x − y | <1, write For the first summand, observe that . For the second summand, an easy calculation shows that, for α ≠ 1, Combining with the estimate for the first summand, and with the case |x − y | ≥1, we obtain that for 0 < α < 1, while for α > 1, Our third example is a variation of our second example above. As in the second example, let X be a finite measure space with measure dx and let W = ð0, 1Þ × X, with dw = s α−1 dsdx, where α > 0. For w = ðs, uÞ ∈ W, let h w ðxÞ = a s ðx, uÞ/ka s ð·, uÞk L 2 . Clearly, kh w k L 2 = 1.
Then, our new distance is given by (For a related example, see Section 4 of [23] and the very last example in Section 2 of [5].) Let us specialize to the case of a symmetric diffusion semigroup with the following additional requirement: a s ðz, zÞ is constant over z ∈ X (but varies with s). Let a s ð·, · Þ denote the value of a s ðz, zÞ for every z ∈ X. Under these assumptions, using the semigroup property, we easily obtain In the very special subcase of X = ℝ n equipped with Lebesgue measure and fA t g the heat flow semigroup, and we thus obtain the same estimates for D 3 ðx, yÞ as for D 2 ðx, yÞ above. We now return to the general development. The following simple result is the key tautology to prove our Theorem 6.
Proof. Using Fubini's theorem and the assumption that a t ðx, yÞ integrates to 1 in each variable, we see that For D ∈ Δ, letting g D ðtÞ ≡ ∬ a t ðx, yÞD 2 ðx, yÞdxdy, we obtain the following result.

Abstract and Applied Analysis
Proof. Using the definition of Dðx, yÞ, Fubini's theorem, and Proposition 3, we observe that Corollary 5. g D ðtÞ < ∞ for every t ∈ T and g D ðtÞ ⟶ 0 as t ⟶ 0.
Proof. Since jhh w , h w − A t h w ij ≤ 2 and Ð W dw < ∞, the result that g D ðtÞ < ∞, for every t ∈ T, follows from Proposition 4. From one of our initial assumptions that for every f ∈ L 2 ðXÞ, kA t f − f k L 2 ⟶ 0, as t ⟶ 0, we obtain that hh w , h w − A t h w i ⟶ 0, as t ⟶ 0, for every w. Hence, g D ðtÞ ⟶ 0, as t ⟶ 0, by the dominated convergence theorem.
Recalling Definition 2, we can now prove the following theorem, which is of interest only due to Corollary 5. Proof. First, suppose that f ∈ L 2 ðXÞ is L 2 -Lipschitz. Then, by Proposition 3, we have for t ∈ T.
, for every t ∈ T . Then, by Proposition 3 again, Thus, f is L 2 -Lipschitz.
It is easy to see that ð1/2Þkf − A t f k 2 so Theorem 6 establishes an equivalence between f being L 2 -Lipschitz and having an estimate of the speed of convergence of k f − A t f k L 2 to 0, as t → 0.
Note that if fA t g is a symmetric semigroup (and T = ½0, ∞Þ), then ð16Þ

Conclusions
For X a topological space equipped with a complete positive σ-finite measure, W a finite measure space, and selecting everywhere-defined real-valued h w ∈ L 2 ðXÞ for every w ∈ W with kh w k L 2 ðXÞ ≤ 1, we have defined a distance D by Dðx, yÞ ≜ kh w ðxÞ − h w ðyÞk L 2 ðWÞ . For T a subset of the reals having 0 as an accumulation point and for t ∈ T, letting a t ðx, yÞ be a nonnegative measurable function on X × X which integrates to 1 in each variable, we have considered bounded operators A t on L 2 ðXÞ given by A t f ðxÞ = Ð X a t ðx, yÞf ðyÞ dy. Assuming that for every f ∈ L 2 ðX Þ, kA t f − f k L 2 ⟶ 0, as t ⟶ 0, t ∈ T, we have shown that ∬ a t ðx, yÞð f ðxÞ − f ðyÞÞ 2 dxdy ≤ c∬ a t ðx, yÞD 2 ðx, yÞ dxdy, for every t ∈ T if and only if ð0 ≤ Þh f , f − A t f i ≤ cg D ðtÞ, where g D ðtÞ = 2 Ð W hh w , h w − A t h w i dw ⟶ 0, as t ⟶ 0, t ∈ T.

Data Availability
No data were used to support this study.

Disclosure
Ramapo College had no involvement with the writing or submission for publication of this work.