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Kenpachi
Kenpachi
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I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

Questions:

  1. I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?

2 What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

Edit: Question 3. Please also advise a little about non-quotient spaces?

Edit:Joonas Ilmavirta answers Q1 and 3 below. Someone please look at Question 2.

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

Questions:

  1. I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?

2 What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

Edit: Question 3. Please also advise a little about non-quotient spaces?

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

Questions:

  1. I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?

2 What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

Edit: Question 3. Please also advise a little about non-quotient spaces?

Edit:Joonas Ilmavirta answers Q1 and 3 below. Someone please look at Question 2.

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Kenpachi
Kenpachi
  • 31
  • 2

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?Questions:

  1. I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?

2 What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

Edit: Question 3. Please also advise a little about non-quotient spaces?

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus? What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

Questions:

  1. I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?

2 What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

Edit: Question 3. Please also advise a little about non-quotient spaces?

Source Link
Kenpachi
Kenpachi
  • 31
  • 2

BMO spaces on the torus

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus? What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.