Riemannian geometry: when is this differential form harmonic?

Leave $ (M ^ 3, g) $ be a Riemannian variety (closed) and leave $ u: M a S $ be a harmonic function where $ S $ It is a closed steerable surface. Yes $ omega $ is a $ 2 $-form in $ S $What are the sufficient conditions for $ omega $ in order $ u ^ * omega $ be a harmonic $ 2 $-form in $ M $?

The specific case I am analyzing is the following: we have $ u_1, u_2: M to mathbb {S} ^ 1 $ harmonic functions, $ u = (u_1, u_2): M to mathbb {S} ^ 1 times mathbb {S} ^ 1 $ Y $ omega = d theta_1 wedge d theta_2 $, where $ d theta_i $ denotes the shape of the volume in each $ mathbb {S} ^ 1 $.

harmonic analysis – classical singular integral operator

I am working on a problem related to the Biot-Savart law in fluid dynamics. I found a singular integral theorem that is closely related to my research.

Assume that $ K (x) $ is a Calderón-Zygmund nucleus in $ mathbb {R} ^ {3} $ Y $ f $ it's a satisfying smooth function
begin {equation}
| f (x) | leq frac {1} {(1 + r) ^ {a}} quad text {para} quad 0 <a <2,
end {equation}

Y
begin {equation}
| nabla f (x) | leq frac {1} {(1 + r) ^ {b}} quad text {para} quad 0 <b <2,
end {equation}

where $ r = sqrt {x_ {1} ^ {2} + x_ {2} ^ {2}} $. It decomposes only in horizontal directions.

Define $ Tf (x) = int K (x-y) f (y) dy $. So there is a constant $ c_ {0} $ such that
begin {equation}
| Tf (x) | leq frac {c_ {0}} {(1 + r) ^ {d}},
end {equation}

where $ d = min {a, b } $.

My questions:
1. Someone suggested to me that this theorem could be improved. Since it is a point or $ L ^ { infty} $ estimate the singular integral that was not covered by any existing theory. Is it possible to demonstrate something like this? $ d = frac {a + b} {2} $ which will help my project a lot.

  1. One can change the second condition or add something new like decomposition of $ nabla ^ {2} f $.

I am not an expert in harmonic analysis. Any advice is welcome. Thank you!

Solution check: is it valid proof that the harmonic series diverges?

Is this valid proof that the harmonic series diverges?

  1. Suppose the series converges to a value, S:

$$ S = 1 + frac {1} {2} + frac {1} {3} + frac {1} {4} + frac {1} {5} + … $$

  1. Divide the series in two, with alternate and odd denominators. As the original series converges, the component series will converge.

$$ S_ {EVEN}} = frac {1} {2} + frac {1} {4} + frac {1} {6} + frac {1} {8} + … $$
$$ S_ {ODD} = 1 + frac {1} {3} + frac {1} {5} + frac {1} {7} + … $$
$$ S = S_ {EVEN}} S_ {ODD} $$

  1. Show that $ S_ {EVEN}} = frac {1} {2} S $

$$ frac {1} {2} S = frac {1} {2} (1+ frac {1} {2} + frac {1} {3} + frac {1} {4} + frac {1} {5} + …) = frac {1} {2} + frac {1} {4} + frac {1} {6} + frac {1} {8} + … = S_ {EVEN}} $$

  1. to show $ S_ {ODD}> S_ {EVEN}} $ because each odd term is greater than its corresponding even term:
    $$ 1> frac {1} {2} qquad frac {1} {3}> frac {1} {4} qquad frac {1} {5}> frac {1} {6} qquad … $$

  2. to show $ S_ {ODD} = S_ {EVEN}} $
    $$ S_ {ODD} = S-S_ {EVEN} = S- frac {1} {2} S = frac {1} {2} S = S_ {EVEN} $$

  3. The contradiction implies that the original assumption of convergence is false:

$$ S_ {ODD}> S_ {EVEN}} $$
$$ S_ {ODD} = S_ {EVEN}} $$
$$ therefore S ne 1+ frac {1} {2} + frac {1} {3} + frac {1} {4} + frac {1} {5} + … $$

co.combinatorics – Slices vs Harmonic functions – intuition / definitions question

You can identify functions in sectors of the Boolean hypercube ($ (n) choose d $ with multilinear harmonic functions of maximum degree d.

I really don't understand the intuition behind the harmonic condition, for everyone $ x_i $ , $$ partial P / partial x_i = 0 $$ and also has some problems with the definitions. I am using the notation as in https://arxiv.org/abs/1406.0142.

It allows focous in case d = 2.
It seems to suggest then if $ P = A_ {ij} x_i x_j $ then A is antisymmetric. But that means that A = 0.
for example ,
$$ (a_ {12} + a_ {21}) x_2 = – (a_ {13} + a_ {31}) x_3 $$
but $ x_2 $ Y $ x_3 $ They are independent, so we can choose what we want.

It seems that, on the one hand, x_i determines if i is in the segment.
Then the functional $< e_{{i,j}} , cdot >$ it translates into $ x_i x_j $ .
On the other hand, $ x_i x_j $ It is not in space.
However, we have a natural transformation B a $ chi_B $ for each top set, but not for the set $ {1,2 } $.

Thank you.

harmonic analysis – Function theory from $ mathbb {Z} _ {p} $ to $ mathbb {Z} _ {q} $ for different cousins ​​$ p, q $

Leave $ p $ Y $ q $ be prime numbers When $ p = q $, Mahler's Theorem gives a complete description of $ C left ( mathbb {Z} _ {p}; mathbb {Z} _ {p} right) $, the space of continuous functions of $ mathbb {Z} p to $ mathbb {Z} p. I wonder (possibly in vain) if there could be a comparable classification of $ C left ( mathbb {Z} _ {p}; mathbb {Z} _ {q} right) $ when $ p $ Y $ q $ They are different

I ask just because I've been doing $ p $-Adical harmonic analysis, but I found myself having to challenge the savages of $ L ^ { infty} left ( mathbb {Z} _ {p}; mathbb {C} _ {q} right) $, everyone's space $ f: mathbb {Z} _ {p} rightarrow mathbb {C} _ {q} $ so that:$$ sup _ { mathfrak {z} in mathbb {Z} _ {p}} left | f left ( mathfrak {z} right) right | _ {q} < infty $$

The duality of Pontryagin allows me to do Fourier analysis in $ L ^ { infty} left ( mathbb {Z} _ {p}; mathbb {C} right) $; for $ p = q $, on the other hand, I can use things like the integral volkenborn, or the amice / mazur-mellin transform—$ p $-Adic distributions, in general. The problem is that, without a structure theorem like Mahler for $ p neq q $ case, although I can define "integration" in $ L ^ { infty} left ( mathbb {Z} _ {p}; mathbb {C} _ {q} right) $ by elements of its dual space (continuous functions $ varphi: L ^ { infty} left ( mathbb {Z} _ {p}; mathbb {C} _ {q} right) rightarrow mathbb {C} _ {q}) $, I don't see a way to make useful calculations for specific non-abstract functions that I am trying to analyze with Fourier.

Any ideas? Referral recommendations? Etc.?

dg. differential geometry: are the harmonic functions in hyperbolic collectors with finite volume constant?

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integration: is the harmonic mean of linear functions a Bernstein function?

According to some experiments I've been running, for any $ n $ and not negative $ a_1, a_2, ldots a_n $, the following function:

$ f (t) = frac {n} { sum_ {i = 1} ^ n 1 / (a_i + t)} $

It is a Bernstein function, which means that its derivatives 1, 3, 5, … are positive and its derivatives 2, 4, … are negative. (Equivalently, its derivative is a completely monotonous function).

The closest quote in the literature I could find is: https://link.springer.com/article/10.1007/s40590-016-0085-y?shared-article-renderer#Sec10, which tests it for 2 variables. Is this true or known in the general case of $ n $ such variables?

differential equations – Change the general solution form of the simple harmonic oscillator

When I try to solve the simple harmonic oscillator equation

DSolve[y''[x] + k*y[x] == 0, y[x], x]

The solution is given by:

{{y[x] -> C[1] Cos[Sqrt[k] x] + C[2] Sin[Sqrt[k] x]}}

How can I get the general solution on the form?

{{y[x] -> C[1] Cos[Sqrt[k] x + C[2]]}}

Hodge's theorem for the cohomology group in holomorphic vector packages and harmonic forms.

Consider the holomorphic vector pack $ pi: E rightarrow M $ where $ M $ it is a complex complex of dimension $ dim _ { mathbb {C}} M = m $. We denote by $ Omega ^ p, q (M) $ the package for bidegree $ p, q $ holomorphic and antiholomorphic forms. The complex star of Hodge is given as $ bar { ast}: Omega ^ {p, q} (M) rightarrow Omega ^ m-p, m-q} (M) $. Now consider the case when we have $ E $ rated antiholomorphic $ q $ way then the star hodge operator is $ ast_ {E}: Omega ^ {0, q} (M) bigotimes E rightarrow Omega ^ {n, nq} bigotimes E ^ {*} = Omega ^ {0, nq} bigotimes K_ {M} bigotimes E ^ {*} $ where $ E * is the dual package and $ K_ {M} = Omega ^ n, 0} (M) $ It is the canonical line package.
We denote by, $ A p, q package section space $ Omega ^ p, q (M) $.
Defining an operator, $ bar { partial} E: A 0, q (E) → A 0, q-1 (E) $ and it's dual as $ bar { partial} ^ {*} _ {E} = (- 1) ^ {q} ast ^ {- 1} _ {E} bar { partial} _ {K_ {M} bigotimes E ^ {*}} ast_ {E} $ where $ A ^ {0, nq} (K_ {M} bigotimes E ^ {*}) xrightarrow { bar { partial} ^ {*} _ {K_ {M} bigotimes E ^ {*}} A 0, nq-1} (K_ {M} bigotimes E ^ *) $.
Now the Laplacian $ Delta_ {E} = bar { partial_ {E}} bar { partial ^ {*} _ {E} +} bar { partial ^ {*} _ {E}} bar { partial_ {E}} $. The core set of this Laplacian operator is harmonic $ (0, q) $ shapes with values ​​in $ E $.
According to Hodge's theorem, the $ k $ The cohomology group of a complex variety is isomorphic to the space of $ k $ degrees of harmonic forms. How does this apply in the present case for holomorphic vector packages. Should we think of the Dolbeault cohomology group? Or free local sheaves?
What group of cohomology is isomorphic to the space of such harmonic forms?

group theory gr. Equivalence of harmonic measures in hyperbolic groups: an elementary test?

Consider a hyperbolic group Gromov $ Gamma $ and let $ mu $ be a measure of probability finely supported by $ Gamma $. Suppose the support of $ mu $ generate $ Gamma $ as a semi-group, in other words, the random walk $ X_n $ conducted by $ mu $ you can visit the whole group $ Gamma $.

Fact: the random walk $ X_n $ almost certainly converges to a point on the limit of Gromov $ partial X $ from $ X $. Leave $ nu $ be the output measure in $ partial X $. So, $ ( partial X, nu) $ It is a model for the so-called Poisson limit. Measure $ nu $ it is called the harmonic measure (with respect to $ mu $)

Now consider the inverse measure $ check { mu} $ defined by $ check { mu} (g) = mu (g -1) $ and let $ check { nu} $ be the corresponding harmonic measure.

Question : They are $ nu $ Y $ check { nu} $ equivalent?

It seems that the answer is positive. In the document Harmonic measures versus quasi-formal measures for hyperbolic groups, Blachère, Haïssinsky and Mathieu demonstrate that $ h = lv $ in a hyperbolic group if and only if the harmonic measure $ nu $ and the Patterson-Sullivan measure $ rho $ are equivalent here, $ h $, is asymptotic entropy, $ l $ is asymptotic drift and $ v $ It is the growth of the volume. Very roughly, the test is like that.

First, both measures are ergodic, so they are equivalent or mutually unique. The Hausdorff dimension of $ nu $ is $ h / l $ and the Hausdorff dimension of $ rho $ is $ v $. Using Lebesgue's differentiation theorem, they show that they are equivalent if and only if they have the same Hausdorff dimension. This part is very technical, the basic ingredients being the shadow slogans for the Patterson-Sullivan measurement and for the harmonic measurement.

From shadow slogans to harmonic measurements $ nu $ Y $ check { nu} $ wait and from $ nu $ Y $ check { nu} $ they have the same Hausdorff dimension, I think you can adapt your test to show that $ nu $ Y $ check { nu} $ they are equivalent

However, I would like to try a similar result in a wider environment and this strategy seems too complicated and too technical. Then I wonder if there is a much simpler test.

Keep in mind that one really needs to use hyperbolicity somewhere. For example, consider a non-centered probability measure $ mu $ in $ mathbb {Z} ^ d $, that is to say
$$ p = sum_ {x in mathbb {Z} ^ d} x mu (x) neq 0. $$
So, $ p $ It is called the drift of the random walk. According to Ney and Spitzer's theorem, Martin's limit of the random walk is a sphere: a sequence of points $ x_n $ converges to a point at the limit if and only if $ x_n $ goes to infinity and $ x_n $ converges in direction, that is $ x_n / | x_n | $ converge The harmonic measure rests in the direction given by $ p $. In particular, the harmonic measure associated with $ check { nu} $ it's compatible with $ -p $ and so $ nu $ Y $ check { nu} $ they are unique